Translation:From inside to outside.
Well, technically, yes. But we would be consistant and use the short form on "dışarı" as well and say: "içten dışa".
No big difference, I guess. But "iç" and "dış" are usable as adjectives meaning "inner" and "outer", like: iç çekirdek → inner core, etc. This is something you can't do with "içeri/dışarı".
When saying inside and outside, usually we use "içeri" and "dışarı".
Yes, that seems to be correct.
One more thing, the dative is optional for "içeri" and "dışarı"
İçeri girmek = İçeriye girmek
Dışarı çıkmak = Dışarıya çıkmak
So, in that case, it's an adverb, obviously, but the lack of the dative suffix might be confusing at first. Just to give you a heads-up.
the root is iç (it means something like internal-hard to translate) and it takes different suffixes for different cases.
inside the house = evin içinde (locative)
come inside = içeriye gir (dative)
it is too hot inside = içerisi çok sıcak (posessed)
there are twenty people inside = içer(i)de yirmi kişi var (locative)
as you can see it all starts with "iç". it is a bit difficult but we don't have many words like this. if i'm not wrong just iç and dış.