yes you can
yes Matt/Flat paint makes a very good undercoat
You will have no problem painting satin over top of flat paint. Just remember to use the appropriate paint that will stick to the old coating. If the old coating is latex (water based) you can go...
If you really want to ... yes. It might cause some issues later but should work for the most part as long as you apply a thin layer of paint.
if they are the same such as both latex, then yes. if one is oil and the other latex then no. if you are not sure apply a universal primer first.