The answer to (1) is to be found in
Rourke, C. P.; Sanderson, B. J.$\Delta$-sets. I. Homotopy theory. Quart. J. Math. Oxford Ser. (2) 22 (1971), 321–338.
It is shown there that a Kan "semi-simplicial" set admits a compatible system of degeneracies.
By the way, the term "semi-simplicial" set is not the usual name for this term; it is usually called a "$\Delta$-set."
Added: Given Mike's comment below, I realize now that the following sketch doesn't do the job.
I haven't looked at this paper recently, but I would imagine the way it goes is as follows: Let $$ \Delta^{\text{inj}} = \text{category of finite ordered sets and order preserving injections} $$
$$ \Delta = \text{category of finite ordered sets and order preserving maps} $$ The we have an inclusion functor $j: \Delta^{\text{inj}} \to \Delta$. Given a "semi-simplicial" set $X$ (i.e., a functor $X: \Delta^{\text{inj}} \to \text{Sets}$) we can form the left Kan extension $j_*X$ and then the restriction $j^*j_*X$ to get a natural map $X \to j^*j_*X$. One can ask whether this is a weak equivalence of simplicial sets. Perhaps what Rourke and Sanderson are doing is showing this map to be a weak homotopy equivalence, or maybe just so when $X$ is Kan? (I don't have the paper at hand, so this is speculation on my part.) One might argue as follows: Step a): show that $j^\ast j_\ast$ preserves colimits, Step b) show that the map $X \to j^\ast j_\ast X$ is a weak equivalence when $X$ is a standard "semi-simplicial" $n$-simplex, Step c) infer the general case by induction on simplices.
At any rate, if this is how it goes, then the outcome also provides an answer to (3).
