"She is asking for a bottle of wine."
Translation:Hun ber om en flaske vin.
"Å stille - stiller" is used specifically as in "asking a question". You can look at it in English as closer to "posing"
She is asking(posing) him a question - hun stiller ham et spørsmål
They are exhibiting(posing, showing off) an artwork - de stiller ut et kunstverk
He is put(posed) before the court - Han er stillt for retten
When you are asking for something as a request you're not posing a question, more as giving a request.
She is asking for a bottle of wine - hun ber om en flaske vin
We ask you to please turn off your cellphones before the show - Vi ber dere om å vennligst skru av mobiltelefonene deres før programmet
As asked in another comment, "spør etter" and "ber etter" are pretty much interchangable.