How much can you do as a homeowner in Texas without requiring a permit/inspections?
For questions one and two:

Legally, it's up to local code. I'm not up on code in Texas. But where I've lived, it's generally acceptable for homeowners to do most work themselves. If it's a structural change (new walls, new foundation, new electrical circuits etc.) It typically requires a permit and inspection.

Granted, just because something requires a permit, it doesn't mean one was pulled. This is common--even if you hire a contractor. Alas, just because you hire a contractor does not mean that a) they will pull permits voluntarily or b) actually do things right anyways.

I've done work as a homeowner with and without permits and I'm still on the fence about it. I appreciate what permits are supposed to do, but have also had to deal with some very backwards and out of date inspectors. I can see why people skip the permit step at times. Sometimes it's to cut corners, but, ironically, I think sometimes it's so they can be left alone to do the job right rather than what the inspector wants to see. ;)

As for question 3, you'll have to seek out a lawyer for that.