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This is exactly the notion of a closed category. See Eilenberg and Kelly's article in the 1965 La Jolla proceedings (Springer 1966). I think they also describe categories enriched in a closed catego …

The following result seems to be well known: If T is a (V-)monad on a (V-)category C, then the forgetful functor $U^T \colon C^T \to C$ creates any limits that exist in C, and any colimits that exis …

I'm presuming that you want to treat V-Cat as a 2-category rather than as a V-category. There might be a slick change-of-base argument that applies here, but I can't think of one. I think the answer …

It is the usual sup metric. See section 2 of Lawvere's original article.

A monoidal functor preserves the unit and tensor only 'laxly', so that $1_X$ is not the local unit but comes with a 2-cell $I \Rightarrow 1_X$. If you're talking about strong monoidal functors then e …