Yes it it kept up to date. Currently 1.6 (newest patch). But no official multiplayer on mobile.
You can, however, copy your mobile save to PC if you have Stardew on PC too. You just have to do the transfer each time (mobile to PC, play on PC, then xfer that PC save back to mobile so the save file stays up to date).
^ This. This is the way (No way am I starting on PC from the beginning as is)