Learn Hardcore Spanish (Spain): ¿Seguimos después de la cena?

Breakdown of ¿Seguimos después de la cena?

What verb form is seguimos, and why is it used in a question like this?

Seguimos is present indicative, first person plural (we) of seguir.
In Spanish, the present can ask about near-future plans and also make a gentle suggestion. So ¿Seguimos…? functions like a soft proposal to continue.

Is seguir irregular? Does seguimos have a stem change?

Seguir is irregular (e→i) in most present forms: sigo, sigues, sigue, seguimos, seguís, siguen.
In the nosotros form (seguimos) there is no stem change, which is why it looks regular here.

What’s the difference between ¿Seguimos…? and Sigamos…?

¿Seguimos…? (present indicative) = a polite question/suggestion: Are we continuing…?
Sigamos… (first person plural imperative, i.e., present subjunctive) = a direct suggestion/encouragement: Let’s continue.
Both are fine; ¿Seguimos…? sounds a bit softer.

Could I say ¿Seguiremos después de la cena? instead? What changes?