How can I clean and smooth the finish on an old table?
Those questions are all interrelated ;)  Whether mineral spirits are safe to use depends on the finish. Any sense of age? Since there's more than one, I'll assume it's factory made and guess a polyurethane finish, and as another guess not oil based since it doesn't appear to have any yellowing... put a drop of water on it.  If it turns white it's waterborne. I'd suggest you try mineral spirits on an inconspicuous spot first, although that won't help much with refinishing  and will often leave a greasy film that may interfere later... you may be better off with naptha.

If you're trying to remove scratches etc, use a fine sandpaper, maybe start with 320 grit before applying a new top coat.  If you're trying for a high gloss, as it appears it has, it's much more difficult by hand... usually sprayed in a factory.  If this is a piece you're very fond of, I'd suggest doing some reading on refinishing and practice on a less important piece first. 

Hope this helps!

Ooops, I see on rereading that you're trying to salvage the finish?  Start with 320 if there are significant scratches, work your way up to 800 or higher, depending on the gloss you're looking for you may want to go as high as 1200.  Stop frequently to clean the slurry from the surface and check your work.  Be especially careful at edges as it's easy to sand through the finish there.