I'm going to assume that $\mathcal{V}$ is closed as well, or at least that its tensor product preserves colimits in each variable; I'm not sure that you get the left adjoint from $\mathsf{Cat}$ to $\mathcal{V}\text{-}\mathsf{Cat}$ otherwise. In this situation the underlying adjunction $\mathsf{Set} \rightleftarrows \mathcal{V}$ is monoidal, i.e. its left adjoint is strong monoidal and its right adjoint lax monoidal. Therefore, the induced monad $S$ on $\mathsf{Set}$ is a monoidal monad and hence the category $S\text{-}\mathsf{Alg}$ is a multicategory. The adjunction on enriched categories is just obtained by applying this adjunction on hom-objects, and so a $T_{\mathcal{V}}$-algebra is just a category enriched over $S\text{-}\mathsf{Alg}$. Thus:
Q1: If $\mathcal{V}$ is monadic over $\mathsf{Set}$ via this adjunction, then so is $\mathcal{V}\text{-}\mathsf{Cat}$ over $\mathsf{Cat}$. For instance, this is the case for $\mathcal{V} = \mathsf{Ab}$.
Q2: It depends a lot on $\mathcal{V}$. For instance, if $\mathcal{V} = \mathsf{Top}$, then $S$ is the identity monad, hence so is $T_{\mathcal{V}}$.
Q3: Because $T_{\mathcal{V}}$ acts as the identity on objects, any lax $T_{\mathcal{V}}$-morphism is automatically a strict $T_{\mathcal{V}}$-morphism, which means in turn that it is homwise an $S$-morphism. Thus, $T_{\mathcal{V}}$ cannot be lax-idempotent unless $S$ is idempotent.
The statement that being additive is more a property than a structure depends on the fact that an additive category is not just an $\mathsf{Ab}$-enriched category, but also has finite products (which are therefore biproducts). This is what enables the enrichment to be characterized in terms of the biproducts, and it makes the forgetful functor from additive categories to ordinary categories pseudomonic, both as a 1-functor and as a 2-functor; thus being additive is a "property-like structure". But I'm not sure if this can be characterized in 2-monadic terms.
Q4: To start with, the fact about additive categories has nothing to do with subtraction, so it is equally true of semiadditive categories, which are enriched only over commutative monoids. Moreover, conversely any category with biproducts is automatically enriched over commutative monoids. In fact, $\mathsf{CMon}$ is the "closed monoidal locally presentable category freely generated by the fact that finite coproducts are absolute in $\mathsf{CMon}$-enriched categories". Other monoidal categories with analogous properties, for which being a $\mathcal{V}$-category with the appropriate kind of colimits is also a property-like structure, include:
- Pointed sets, for initial objects (which are then zero objects)
- Suplattices, for arbitrary coproducts (which are then biproducts)
- In the $\infty$-categorical world, spectra, for arbitrary finite colimits (which then satisfy the stability property that pushout squares coincide with pullback squares).
Moritz Rahn (nee Groth) and I wrote something about this phenomenon in Generalized stability for abstract homotopy theories, and intended to write a sequel, but haven't gotten around to it yet.
