It is possible to build a ~31" AR rifle and remain legal. But that's as short as you can go.
That's minimum legal barrel length, and minimum length required for a carbine buffer system to work (in a pistol-length tube, for that matter). My numbers: A pistol-length carbine buffer tube* and standard AR receiver set locate the case head about 14-3/4" from the end of the buffer tube. Add a 16" barrel, and you're at 30-3/4". But, no one makes 16" barrels; they're all 16.25" or 16.5", just to be sure they're legal. So, you're at 31" (or more).
*(I can't provide a link at this time, but I have seen pistol-length tubes that accept an adjustable stock. I know they're out there. I just can't cite an example right now.)
Any shorter means one of two things:
1. The barrel length is too short, and requires an SBR stamp.
2. You've gone to one of the 9mm pistol buffer systems that are super-short and won't allow a stock to be installed.
Light weight:
I've gone through quite a few TAC-15 lower receivers. They're a reinforced polymer lower with brass inserts. Fairly light. I haven't damaged one. But they're quite bad, dimensionally, and require too much fiddling, fitting, and shimming.
I am not a fan. I wouldn't recommend them.
Unless you go through the background check and paperwork, and pay the stamp fee, then you can not build an SBR. The current wait times are over 6 months I believe.
The Eastern US seems to be stuck at around 6 months; but many Western states have had a big uptick in shorter turn-around over the last couple months. I recently got a Form 1 (SBR) approved in just 3 months, for example (Idaho - 3 months, 5 days, including time in the mail).
I had the receiver engraved, assembled in SBR configuration, and firing, less than a week later.
There are, of course, additional legal considerations for SBRs, though.