In this question $(\mathcal{V}, \otimes, e)$ is a (bi)complete symmetric monoidal category.

We have an adjunction $F: \mathsf{Cat} \leftrightarrows \mathcal{V}\text{-}\mathsf{Cat} :(-)_0$, induced by the change of enrichment as discussed for example between Example 3.2 and 3.3 here. Let me call $\mathsf{T}_{\mathcal{V}}$ the induced monad on $\mathsf{Cat}$.

Q1. Do we know, or can we characterize, when this adjunction is monadic?

Q2. When it is not monadic, how should I think about an algebra for $\mathsf{T}_{\mathcal{V}}$?

Sometimes one reads that being an additive category (i.e. being $\mathsf{Ab}$-enriched) is more a property than a structure, see for example this question which actually motivated the present question, but I am sure you met such statements in your life.

Q3. Does this mean that the monad $\mathsf{T}_{\mathsf{Ab}}$ is lax-idempotent, or it is some different behavior?

Q4. Do you have other examples, or a characterization, or a sufficient condition such that being a $\mathcal{V}$-category is a property, in the same way this is true for additive categories? (I am aware this is not a well-posed question).

In this question $(\mathcal{V}, \otimes, e)$ is a (bi)complete symmetric monoidal category.

We have an adjunction $$\mathscr{l}: \mathsf{Cat} \leftrightarrows \mathcal{V}\text{-}\mathsf{Cat} :(-)_0,$$ induced by the change of enrichment as discussed for example between Ex. 3.2 and 3.3 here. Let me call $\mathsf{T}_{\mathcal{V}}$ the induced monad on $\mathsf{Cat}$.

Q1. Do we know, or can we characterize, when this adjunction is monadic?

Q2. When it is not monadic, how should I think about an algebra for $\mathsf{T}_{\mathcal{V}}$?

Sometimes one reads that being an additive category (i.e. being $\mathsf{Ab}$-enriched) is more a property than a structure, see for example this question which actually motivated the present question, but I am sure you met such statements in your life.

Q3. Does this mean that the monad $\mathsf{T}_{\mathsf{Ab}}$ is lax-idempotent, or it is some different behavior?

Q4. Do you have other examples, or a characterization, or a sufficient condition such that being a $\mathcal{V}$-category is a property, in the same way this is true for additive categories? (I am aware this is not a well-posed question).

In this question $(\mathcal{V}, \otimes, e)$ is a (bi)complete symmetric monoidal category.

We have an adjunction $F: \mathsf{Cat} \leftrightarrows \mathcal{V}\text{-}\mathsf{Cat} :(-)_0$, induced by the change of enrichment as discussed for example between Example 3.2 and 3.3 here. Let me call $\mathsf{T}_{\mathcal{V}}$ the monad on $\mathsf{Cat}$.

Q1. Do we know, or can we characterize, when this adjunction is monadic?

Q2. When it is not monadic, how should I think about an algebra for $\mathsf{T}_{\mathcal{V}}$?

Sometimes one reads that being an additive category (i.e. being $\mathsf{Ab}$-enriched) is more a property than a structure, see for example this question which actually motivated the present question, but I am sure you met such statements in your life.

Q3. Does this mean that the monad $\mathsf{T}_{\mathsf{Ab}}$ is lax-idempotent, or it is some different behavior?

Q4. Do you have other examples, or a characterization, or a sufficient condition such that being a $\mathcal{V}$-category is a property, in the same way this is true for additive categories? (I am aware this is not a well-posed question).

In this question $(\mathcal{V}, \otimes, e)$ is a (bi)complete symmetric monoidal category.

We have an adjunction $F: \mathsf{Cat} \leftrightarrows \mathcal{V}\text{-}\mathsf{Cat} :(-)_0$, induced by the change of enrichment as discussed for example between Example 3.2 and 3.3 here. Let me call $\mathsf{T}_{\mathcal{V}}$ the induced monad on $\mathsf{Cat}$.

Q1. Do we know, or can we characterize, when this adjunction is monadic?

Q2. When it is not monadic, how should I think about an algebra for $\mathsf{T}_{\mathcal{V}}$?

Sometimes one reads that being an additive category (i.e. being $\mathsf{Ab}$-enriched) is more a property than a structure, see for example this question which actually motivated the present question, but I am sure you met such statements in your life.

Q3. Does this mean that the monad $\mathsf{T}_{\mathsf{Ab}}$ is lax-idempotent, or it is some different behavior?

Q4. Do you have other examples, or a characterization, or a sufficient condition such that being a $\mathcal{V}$-category is a property, in the same way this is true for additive categories? (I am aware this is not a well-posed question).