Do Turntables sound better with time?

Of course. Motor bearings and things relax and bed-in with time. The same way you can't use launch control in a Dodge Demon until you've done 500 miles. 

Depends more on the repeatability of the owner's skill in setting up a phono cartridge.  Also, all audio systems sound different on different days, for many different reasons. Ask a better question.

I have never observed that. Doesn’t mean it doesn’t happen… but if it does it is pretty subtle.

The turntable is more likely to be a constant. The phono cartridge is likely to improve with playing time as the suspension of the cantilever loosens up.  I suppose it's possible that the preamp or amp were "burning in" and improving with time, if they were new.  

NO.  Maybe the cartridge, but in my experience (55 years since my 1st turntable) a turntable will not change, unless it starts to breakdown. As noted above, other factors may improve the sound.