Semi gloss or satin is easier to keep clean.

You can clean the walls with tsp before you paint, mostly where greasy hands marked up the wall. Home depot has a newer tsp type product that doesn't need to be washed off. It's a pink liquid in a clear bottle.

If you dont use tsp at least spray some kitchen degreaser around the light switches n doorways to get the schmutz off.

If you do any drywall work the raw drywall and drywall mudd needs to be primed with a primer before painting. Even if the paint says primer n paint in one. I was told to always prime first.

Yeah go right over the old flat paint with whatever you choose no need to prime over paint unless its smoke stained or something.

Flat is not a color. It refers to gloss or sheen, or more specifically a lack there of. Flat paint is the least durable. That’s why it is only used on ceilings. You should at least use a satin.

If you don’t really care what color white it is ask for the untinted paint base. The color of the untinted paint base will vary from manufacturer to manufacturer and also over time as they are using the cheapest crap they can for the base. Tthe untainted base may be cheaper as it is the pigments that are the most expensive component of the paint. All of the paint manufacturers colors that you see on their website are purposely designed to be mixed with affordable pigments. Getting a custom color match can double the price of a gallon of paint sometimes