sondern is only employed after a negation in the antecedent.

https://yourdailygerman.com/meaning-of-sondern/

Aber cannot be used for mutually exclusive alternatives and if there is no possibility to have none of the options:

• Ich habe kein Bier, aber (ich habe) Wein. (I could as well have no drinks at all, or both)
• Heute ist nicht Freitag, sondern Dienstag. (Today must be one of the days of the week, there is no none-day, and no two days at once)

Doch means yet. And the implication with it is that its agreeing with the first clause and continuing to introduce the next clause. "But" (sondern/aber) implies a difference between the two clauses. ~I hope that made sense.

'Sein' leaves things that the object 'is' as nominative. 'Einen' would be the accusative form.

Keine Frau. It's always kein when you would use an indefinite article.

it's because after the verb sein you use nominative rather than accusative