I am painting an old chest that How many coats of paint should I use?

The number of coats depends on the paint color and either flat like chalk paint or shiny like a high gloss. You will have to play it as you go. Good luck

I have opened up living room and dinning area to look into island kitchen. Walls now are light beige/yellowish. When it was painted the trim was painted with enamel over enamel and and the doors,

it is all peeling. What is the best thing to do with the whole room?

I would sand it first then use a good paint with primer already in it. You may only need one coat so purchase a quart. If you are worried ask an independent paint store owner

It depends on what the trunk is covered with. Is it leather, wood, plastic, metal?

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Not knowing what you started with, I would do a light sanding, at least one coat of a good primer for the surface you are painting - wood - metal - plastic, then a couple coats of your desired color.

I think this chair got one coat of paint. But it would have been oil paint. I only wash the furniture, no sanding or prime coat but I am speaking of wood only.