They do make paint especially for ceilings. Or flat for ceiling and eggshell/satin for trim.

It's personal preference. I painted my wood ceiling using gloss because I have low ceilings and wanted to bounce light. I would probably use satin on higher ceilings.

A lower gloss (satin or eggshell of flat) is normally the way to go on ceilings. As Lila points out, the gloss is used only to bounce light into a low ceiling room.

Personally I would just white wash the wood in a matte... two or three parts paint one part water brushed on not rolled then wiped off... I mean that is an expensive cool feature to just cover with solid opaque boring old white...

When I painted mine the first coat looked like whitewashing and it was beautiful. Unfortunately the previous owners had painted the adjoining room so I had to match it. Why not give Tracy's suggestion a try. You can always paint extra coats if you don't like it.