Monomorphisms, epimorphisms, (co-)images and factorizations in $\infty$-categories

An $n$-monomorphism is a map $A\to B$ for which $Map(X,A)\to Map(X,B)$ has all homotopy fibers $n$-truncated, for all $X$; a space (=$\infty$-groupoid) is $n$-truncated if its homotopy groups all vanish in dimensions $>n$.

An $n$-epimorphism is an $n$-monomorphism in the opposite $\infty$-category.

When $n=-1$ you recover Denis's definition of monomorphism and epimorphism: note that a space is $(-1)$-truncated if and only if it is empty or contractible.

The phenomenon of "$n$-epi/mono factorization" in an $\infty$-category does not actually involve factorization as ($n$-mono)($n$-epi), but rather as ($n$-mono)($n$-connected map). The class of $n$-connected maps is usually defined as a kind of left complement to $n$-monomorphisms, so the existence of such factorizations is basically the assertion that your $\infty$-category admits a factorization system where the right-class is the class of $n$-monos. (These exist in all "reasonable" examples, e.g., presentable $\infty$-categories.)

This same issue shows up in 1-category theory: epi/mono factorization tends to be rare, and more commonly one considers factorizations of the form: (regular epi)(mono).

The factorization ($n$-mono)($n$-connected) can be thought of as factorization through an "$n$-image". It is clearly not a self-dual notion, hence there is a dual but distinct notion of factorization through an "$n$-coimage".

I can answer to 4: https://arxiv.org/abs/1501.04658 here the definition is given for a generic poset $J$, provided it has a monotone $\mathbb{Z}$-action.

For 1,2,3 and the preamble, I think that the theory of factorization systems is pretty much unchanged from 1-category theory: see here; the same theorems hold, provided you translate them into quasicategorical terms (this is phrased in a specific model, and yet it is maybe possible to write it up model-independently -doing so in a few specific examples is part of my current plans :-) ).

What changes, when you upgrade to $(\infty,1)$-categories, is the fact that $(Epi, Mono)$ on $\bf Set$ is no longer the most natural example, replaced by $(n\text{-}Epi, n\text{-}Mono)$ on $\infty\text{-}\bf Gpd$.

Notice that all I'm saying is true for $(\infty\color{red}{,1})$-categories only, not $(\infty,n)$-categories, not $(\infty,\infty)$-categories.