Your question (1) is pretty classical -- the Birman Hilden Annals paper from 1973 essentially answers it, no? I mean, they don't deal with the non-orientable case in their paper as far as I recall but the techniques still work.
Re (3), your notion of mapping class group for a surface with boundary would typically be called the mapping class group of a closed surface with marked points, in the literature I'm familiar with, denoted something like $MCG(X,n)$ for a closed surface $X$ and $n$ marked points -- in your case they would be circle boundary components.
In that regard there are extensions ($X$ a boundaryless surface)
$pi_1 Diff(X) to pi_1 C_n X to MCG(X,n) to MCG(X) to 0$
$C_n(X)$ is the configuration space of $n$ points in the surface $X$.
$pi_1 Diff(X)$ is typically trivial but there are some non-trivial cases like the torus, sphere $mathbb RP^2$, the Klein bottle. Some of these special cases give you injectivity $MCG(X,n) to MCG(X)$ for $n$ small, but not many.
Anyhow, it's a natural map and is frequently useful. Are you really more interested in knowing exactly when there is such an embedding or not, or are you more interested in general relations? ie: do you have a reason for wanting to know the answer to these questions?
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