Después de cenar, me quedo un rato en el sofá y escucho música.

Because after después de (after), Spanish uses:

• de + infinitive when the next action is a verb: Después de cenar = After having dinner / After dinner.
  You don’t use a conjugated verb (ceno) directly after de.

Yes. Both are natural:

• Después de cenar = after eating dinner (focus on the action)
• Después de la cena = after the dinner (focus on the event/meal)
  In everyday Spanish, Después de cenar is very common.

It comes from quedarse (to stay / to remain).
me quedo = I stay / I remain (for a while).
In this context it means you don’t go anywhere—you stay on the sofa for some time.

Quedar and quedarse are different:

• quedar often means to meet / to be left / to suit (depends on context): Quedamos mañana (We’ll meet tomorrow).
• quedarse means to stay / to remain: Me quedo en casa (I’m staying at home).
  Here you need quedarse because it’s about staying.

un rato means a while (an unspecified short-to-medium time).
It’s intentionally vague—could be 10 minutes or an hour, depending on context.

In Spanish, when you mean you are sitting/lying and staying on something like a sofa, bed, chair, etc., you typically use en:

• en el sofá = on the sofa
  sobre is more like on top of / above, and can sound odd here unless you’re emphasizing physical position in a special way.

Because Después de cenar is an introductory time phrase. Spanish often uses a comma after long or fronted introductory phrases:

• Después de cenar, me quedo…
  It’s good style and very common.

It can be either, but in a sentence like this it very commonly expresses a routine:

• After dinner, I stay… and I listen to music (habit)
  Spanish present tense often covers habitual actions without needing usually.

Yes, that’s also very natural:

• me quedo… escuchando música = I stay… listening to music
  Using y escucho música separates the actions more; using escuchando highlights that listening happens during the time you’re staying.

Because the subject is already clear from the verb ending:

• escucho = I listen
  Adding yo is optional and usually used for emphasis or contrast (e.g., yo listen, not someone else).