Just clean the existing wall with a damp cloth.

Dont need primer.

1 coat or 2 depends. If you want more ommph to bring out the colour, two coat.

Prep work ie wiping down the wall first is important.
Existing is new paint no need primer*.

*Previous case issue above was due to hacked spot wasn't primed thats why it absorb paint diff from rest-of-wall.

New roller refills are very cheap for the tiny one (toilet paper cardboard size). The big one harder to control if you not painter by profession.
Important to buy a roller handle that the stick is centered properly vs the roller - otherwise your hand also tired AF for nothing if imbalanced.

As for 1 coat or 2 coat if not too fussy and since its off-white on white it may work for you. If paint strong/bright colours I prefer 2 coats. Sometimes wall not even may miss a spot.

paint one layer first to see if the painted area can be covered properly or not, most of the time there will be some spots no covered, or some dirt get into the paint and you need sanding/scraping then 2nd layer can cover that as well as some imperfection from the base and previous layer

if the wall is new and the paint quality is good and not powdery then no need primer, just clean with damp cloth, if you noticed the damp cloth got white powder after you wipe, maybe need to check if they added powder to cheat the paint volume