#1
I'm still a beginner, but I was wondering if it's smart to start working in fingerstyle to my playing?

So many songs that I like are fingerstyle. I have basic chords down pretty well and barre chords are coming in. Should I work on some basic fingerstyle stuff as well as everything else?
#2
It depends on what your goals are and if you have the time to practice it. I can't see any disadvantages with it, except you get less time to practice your "main" techniques with flatpick (I assume?). Check out basic travis picking patterns to start off with! Or just trying to pluck the strings with your fingers if you not have that down yet.
#3
If you want to learn fingerpicking then it is definitely not too early if you know your basic chords and are learning barre chords. I think I learned my first fingerpicking pattern 3 months in and later on I just got hooked.

I am a little over 8 months in now and I can fingerpick many songs that I would never have thought I would be able to do this early.