Color sanding and polishing was invented on lacquer paint. That is how you get a great lacquer finish. If the lacquer is well cured it should take a urethane clear coat just fine, just sand it w/ 400 before the clear. On a small piece like that just color sand it w/ 1500/2000 grit.
Clear coats tent to be quite glossy. It would be highly recommended to do at least a quick sand on the existing surface to create a rough surface to grab. It would be best to completely sand down first though or you may have discoloration issues,
It is recommended to sand, or at least scuff it up so that it has a profile. A primer needs a profile to bond properly with the item you are refinishing.
Primer paint can pretty much be used over top of any other types of paint. It depends on the material of the chair you are painting though. It is always recommended, however, that you remove as much old paint as possible this way you will not have to worry about possible bubbles or flaking.