I will try to give an answer to your first question.

The functor $\Delta_F$ verifies automatically nearly all the conditions of the monadicity theorem:

• it is a right adjoint;
• it is a left adjoint with cocomplete domain, and thus coequalizers exist in the source and are preserved by $\Delta_F$.

It remains to see when the functor $\Delta_F$ is conservative, i.e. reflects isomorphisms. This is true, for example, in each of the following two cases:

• $F$ is essentially surjective;
• $\Delta_F$ is full and faithful. This condition holds, for example, if $F$ induces an equivalence of Cauchy completions, or $F$ is a reflection onto a reflective subcategory, or $F$ is the result of formally inverting some arrows (such as the map from a model category to its homotopy category).

I will try to give an answer to your first question.

The functor $\Delta_F$ verifies automatically nearly all the conditions of the monadicity theorem:

• it is a right adjoint;
• it is a left adjoint with cocomplete domain, and thus coequalizers exist in the source and are preserved by $\Delta_F$.

It remains to see when the functor $\Delta_F$ is conservative, i.e. reflects isomorphisms. This is true, for example, in each of the following two cases:

• $F$ is essentially surjective;
• $\Delta_F$ is full and faithful. This condition holds if $F$ induces an equivalence of Cauchy completions, i.e. $F$ is full and faithful and every object in the codomain is a retract of an object in the image. $\Delta_F$ is also full and faithful when $F$ is a reflection onto a reflective subcategory, or more generally when $F$ formally inverts some arrows (such as the map from a model category to its homotopy category).