It is always a model category using Kan's theory of Reedy categories. For a proof, see Hirschhorn Model Categories and their Localizations 15.3.

This is because $\Delta$ and $\Delta^{op}$ are both Reedy categories.

I will address the more general question as well:

If $C$ is any small category, the condition for $M^C$ to be a model category is that $M$ be cofibrantly generated. However, it is not necessarily true that $M^C$ is itself cofibrantly generated. In general, the condition we need for that is called combinatoriality. However, the existence of such a cofibrantly generated model structure that is not combinatorial would disprove Vopenka's principle (a large cardinal axiom), so I cannot give an example of one.

It always has a model structure using Kan's theory of Reedy categories. For a proof, see Hirschhorn Model Categories and their Localizations 15.3.

This is because $\Delta$ and $\Delta^{op}$ are both Reedy categories.

I will address the more general question as well:

If $C$ is any small category, the condition for $M^C$ to be a model category is that $M$ be cofibrantly generated. However, it is not necessarily true that $M^C$ is itself cofibrantly generated. In general, the condition we need for that is called combinatoriality. However, the existence of such a cofibrantly generated model structure that is not combinatorial would disprove Vopenka's principle (a large cardinal axiom), so I cannot give an example of one.

It is always a model category using Kan's theory of Reedy categories. For a proof, see Hirschhorn Model Categories and their Localizations 15.3.

This is because $\Delta$ and $\Delta^{op}$ are both Reedy categories.

It is always a model category using Kan's theory of Reedy categories. For a proof, see Hirschhorn Model Categories and their Localizations 15.3.

This is because $\Delta$ and $\Delta^{op}$ are both Reedy categories.

I will address the more general question as well:

If $C$ is any small category, the condition for $M^C$ to be a model category is that $M$ be cofibrantly generated. However, it is not necessarily true that $M^C$ is itself cofibrantly generated. In general, the condition we need for that is called combinatoriality. However, the existence of such a cofibrantly generated model structure that is not combinatorial would disprove Vopenka's principle (a large cardinal axiom), so I cannot give an example of one.