The heart of the diagram chase is: All the operations and equations involved in an algebra structure on X have target X. (More generally: the target of each operation/equation could be any limit-preserving functor of the algebra-specified-so-far.)

For each of the kinds of “algebra” you describe, the structure is built up by sequentially adding operations and equations.

Each “add an operation” step fits into the template of taking an inserter. You have two categories and two parallel functors $\newcommand{\C}{\mathbf{C}}\newcommand{\D}{\mathbf{D}}F, G : \C \to \D$; then the inserter category $\newcommand{\Ins}{\mathrm{Ins}}\Ins(F,G)$ consists of objects $X$ of $\C$ equipped with a map $x : FX \to GX$, and a map $f:(X,x) \to (Y,y)$ is a map $f:X \to Y$ in $\C$ satisfying $(Gf)x = y(Ff)$.

(If $T$ is a monad on $\newcommand{\E}{\mathcal{E}}\E$, then the first stage of defining monad algebras is equipping objects with a map $TX \to X$, i.e. taking the inserter of $\Ins(T,1_\E)$.)

Similarly, each “add an equation” stage fits into the template of taking an equifier: you have categories and functors $F,G : \C \to \D$ as before, plus natural transformations $\alpha,\beta : F \to G$, and the equifier $\newcommand{\Eqf}{\mathrm{Eqf}}\Eqf(\alpha,\beta)$ is the full subcategory of $\C$ of objects $X$ for which $\alpha_X = \beta_X$.

(Back with our monad $T$ on $\E$: after adding the operation, getting the category $\Ins(T,1_\E)$ with a forgetful functor $U$ to $\E$, we can now impose the associativity axiom by taking the equifier for the functors $T^2U, U : \Ins(T,1_\E) \to \E$, with the natural transformations sending $(X,x)$ to the maps $x(Tx), x\mu_X : T^2 X \to X$.)

So monad algebras, and their forgetful functor, can be built up as $\newcommand{\Alg}{\mathrm{Alg}}\Alg_\E(T) = \E_3 \to \E_2 \to \E_1 \to \E_0 = \E$, where each step $\E_{n+1} \to \E_n$ “adds an operation or equation”, i.e. is the forgetful functor from an inserter or equifier category. And moreover, “the target of each operation/equation on $X$ is just $X$”, i.e. the target functor $G$ of each inserter/equifier is the composite forgetful functor $\E_n \to \E$. (And algebras for endofunctors or pointed endofunctors can be built up in the same way.)

Now for creation of limits:

Proposition.

1. Given $F,G : \C \to \D$, the forgetful functor $\Ins(F,G) \to \C$ creates all limits that exist in $\C$.

2. Given $\alpha, \beta : F \to G : \C \to \D$, the forgetful functor $\Eqf(\alpha,\beta) \to \C$ creates all limits that exist in $\C$.

The proof of this proposition is that diagram chase you don’t want to do — it has to be done sooner or later. But once it’s done here, it applies to give creation of limits for monad algebras, endofunctors, etc, by their presentation in terms of inserters/equifiers as above.

The heart of the diagram chase is: All the operations and equations involved in an algebra structure on X have target X. (More generally: the target of each operation/equation could be any limit-preserving functor of the algebra-specified-so-far.)

For each of the kinds of “algebra” you describe, the structure is built up by sequentially adding operations and equations.

Each “add an operation” step fits into the template of taking an inserter. You have two categories and two parallel functors $\newcommand{\C}{\mathbf{C}}\newcommand{\D}{\mathbf{D}}F, G : \C \to \D$; then the inserter category $\newcommand{\Ins}{\mathrm{Ins}}\Ins(F,G)$ consists of objects $X$ of $\C$ equipped with a map $x : FX \to GX$, and a map $f:(X,x) \to (Y,y)$ is a map $f:X \to Y$ in $\C$ satisfying $(Gf)x = y(Ff)$.

(If $T$ is a monad on $\newcommand{\E}{\mathcal{E}}\E$, then the first stage of defining monad algebras is equipping objects with a map $TX \to X$, i.e. taking the inserter of $\Ins(T,1_\E)$.)

Similarly, each “add an equation” stage fits into the template of taking an equifier: you have categories and functors $F,G : \C \to \D$ as before, plus natural transformations $\alpha,\beta : F \to G$, and the equifier $\newcommand{\Eqf}{\mathrm{Eqf}}\Eqf(\alpha,\beta)$ is the full subcategory of $\C$ of objects $X$ for which $\alpha_X = \beta_X$.

(Back with our monad $T$ on $\E$: after adding the operation, getting the category $\Ins(T,1_\E)$ with a forgetful functor $U$ to $\E$, we can now impose the associativity axiom by taking the equifier for the functors $T^2U, U : \Ins(T,1_\E) \to \E$, with the natural transformations sending $(X,x)$ to the maps $x(Tx), x\mu_X : T^2 X \to X$.)

So monad algebras, and their forgetful functor, can be built up as $\newcommand{\Alg}{\mathrm{Alg}}\Alg_\E(T) = \E_3 \to \E_2 \to \E_1 \to \E_0 = \E$, where each step $\E_{n+1} \to \E_n$ “adds an operation or equation”, i.e. is the forgetful functor from an inserter or equifier category. And moreover, “the target of each operation/equation on $X$ is just $X$”, i.e. the target functor $G$ of each inserter/equifier is the composite forgetful functor $\E_n \to \E$. (And algebras for endofunctors or pointed endofunctors can be built up in the same way.)

Now for creation of limits:

Proposition.

1. Given $F,G : \C \to \D$, the forgetful functor $\Ins(F,G) \to \C$ creates all limits that exist in $\C$ and are preserved by $G$. Dually, it creates all colimits that exist in $\C$ and are preserved by $F$.

2. Given $\alpha, \beta : F \to G : \C \to \D$, the forgetful functor $\Eqf(\alpha,\beta) \to \C$ creates all limits that exist in $\C$ and are preserved by $G$. Dually, it creates all colimits that exist in $\C$ and are preserved by $F$.

The proof of this proposition is that diagram chase you don’t want to do — it has to be done sooner or later. But once it’s done here, it applies to give creation of limits for monad algebras, endofunctors, etc, by their presentation in terms of inserters/equifiers as above. It also (dually) gives creation of colimits for coalgebras over comonads, endofunctors, etc.; and also directly gives creation of limits/colimits for various other structures, e.g. monoids/comonoids in a monoidal category, without needing to show they’re (co)monadic.

So I think it quite satisfactorily answers your questions (1) and (3). It doesn’t answer your question (2) — I’m surprised to learn that this doesn’t generalise straightforwardly to the enriched setting, and haven’t worked it through enough to understand why — but it should help clarify what happens there, since the decomposition of algebras in terms of inserters and equifiers still holds, even if the proposition about creation of limits fails.

The heart of the diagram chase is: All the operations and equations involved in an algebra structure on X have target X. (More generally: the target of each operation/equation could be any limit-preserving functor of the algebra-specified-so-far.)

For each of the kinds of “algebra” you describe, the structure is built up by sequentially adding operations and equations.

Each “add an operation” step fits into the template of taking an inserter. You have two categories and two parallel functors $\newcommand{\C}{\mathbf{C}}\newcommand{\D}{\mathbf{D}}F, G : \C \to \D$; then the inserter category $\newcommand{\Ins}{\mathrm{Ins}}\Ins(F,G)$ consists of objects $X$ of $\C$ equipped with a map $x : FX \to GX$, and a map $f:(X,x) \to (Y,y)$ is a map $f:X \to Y$ in $\C$ satisfying $(Gf)x = y(Ff)$.

(If $T$ is a monad on $\newcommand{\E}{\mathcal{E}}\E$, then the first stage of defining monad algebras is equipping objects with a map $TX \to X$, i.e. taking the inserter of $\Ins(T,1_\E)$.)

Similarly, each “add an equation” stage fits into the template of taking an equifier: you have categories and functors $F,G : \C \to \D$ as before, plus natural transformations $\alpha,\beta : F \to G$, and the equifier $\newcommand{\Eqf}{\mathrm{Eqf}}\Eqf(\alpha,\beta)$ is the full subcategory of $\C$ of objects $X$ for which $\alpha_X = \beta_X$.

(Back with our monad $T$ on $\E$: after adding the operation, getting the category $\Ins(T,1_\E)$ with a forgetful functor $U$ to $\E$, we can now impose the associativity axiom by taking the equifier for the functors $T^2U, U : \Ins(T,1_\E) \to \E$, with the natural transformations sending $(X,x)$ to the maps $x(Tx), x\mu_X : T^2 X \to X$.)

So monad algebras, and their forgetful functor, can be built up as $\newcommand{\Alg}{\mathrm{Alg}}\Alg_\E(T) = \E_3 \to \E_2 \to \E_1 \to \E_0 = \E$, where each step $\E_{n+1} \to \E_n$ “adds an operation or equation”, i.e. is the forgetful functor from an inserter or equifier category. And moreover, “the target of each operation/equation on $X$ is just $X$”, i.e. the target functor $G$ of each inserter/equifier is the composite forgetful functor $\E_n \to \E$. (And algebras for endofunctors or pointed endofunctors can be built up in the same way.)

Now for creation of limits:

Proposition.

1. Given $F,G : \C \to \D$, if $\C$ has all limits of some shape and $G$ preserves these, then the forgetful functor $\Ins(\C,\D) \to \C$ creates these limits.

2. Given $\alpha, \beta : F \to G : \C \to \D$, if $\C$ has all limits of some shape and $G$ preserves these, then the forgetful functor $\Eqf(\C,\D) \to \C$ creates these limits.

The proof of this proposition is that diagram chase you don’t want to do — it has to be done sooner or later. But once it’s done here, it applies to give creation of limits for monad algebras, endofunctors, etc, by their presentation in terms of inserters/equifiers as above.

The heart of the diagram chase is: All the operations and equations involved in an algebra structure on X have target X. (More generally: the target of each operation/equation could be any limit-preserving functor of the algebra-specified-so-far.)

For each of the kinds of “algebra” you describe, the structure is built up by sequentially adding operations and equations.

Each “add an operation” step fits into the template of taking an inserter. You have two categories and two parallel functors $\newcommand{\C}{\mathbf{C}}\newcommand{\D}{\mathbf{D}}F, G : \C \to \D$; then the inserter category $\newcommand{\Ins}{\mathrm{Ins}}\Ins(F,G)$ consists of objects $X$ of $\C$ equipped with a map $x : FX \to GX$, and a map $f:(X,x) \to (Y,y)$ is a map $f:X \to Y$ in $\C$ satisfying $(Gf)x = y(Ff)$.

(If $T$ is a monad on $\newcommand{\E}{\mathcal{E}}\E$, then the first stage of defining monad algebras is equipping objects with a map $TX \to X$, i.e. taking the inserter of $\Ins(T,1_\E)$.)

Similarly, each “add an equation” stage fits into the template of taking an equifier: you have categories and functors $F,G : \C \to \D$ as before, plus natural transformations $\alpha,\beta : F \to G$, and the equifier $\newcommand{\Eqf}{\mathrm{Eqf}}\Eqf(\alpha,\beta)$ is the full subcategory of $\C$ of objects $X$ for which $\alpha_X = \beta_X$.

(Back with our monad $T$ on $\E$: after adding the operation, getting the category $\Ins(T,1_\E)$ with a forgetful functor $U$ to $\E$, we can now impose the associativity axiom by taking the equifier for the functors $T^2U, U : \Ins(T,1_\E) \to \E$, with the natural transformations sending $(X,x)$ to the maps $x(Tx), x\mu_X : T^2 X \to X$.)

So monad algebras, and their forgetful functor, can be built up as $\newcommand{\Alg}{\mathrm{Alg}}\Alg_\E(T) = \E_3 \to \E_2 \to \E_1 \to \E_0 = \E$, where each step $\E_{n+1} \to \E_n$ “adds an operation or equation”, i.e. is the forgetful functor from an inserter or equifier category. And moreover, “the target of each operation/equation on $X$ is just $X$”, i.e. the target functor $G$ of each inserter/equifier is the composite forgetful functor $\E_n \to \E$. (And algebras for endofunctors or pointed endofunctors can be built up in the same way.)

Now for creation of limits:

Proposition.

1. Given $F,G : \C \to \D$, the forgetful functor $\Ins(F,G) \to \C$ creates all limits that exist in $\C$.

2. Given $\alpha, \beta : F \to G : \C \to \D$, the forgetful functor $\Eqf(\alpha,\beta) \to \C$ creates all limits that exist in $\C$.

The proof of this proposition is that diagram chase you don’t want to do — it has to be done sooner or later. But once it’s done here, it applies to give creation of limits for monad algebras, endofunctors, etc, by their presentation in terms of inserters/equifiers as above.