You can, where the steel bridge crosses the river there is actually a small beach. The park is actually called Schreiner Park. If you go out to google maps and search you can get some pretty good pictures of the area.

Absolutely! It's river swimming at it's best. The Llano River is beautiful.

Yes

Yes our children actually found a rope swing and it was a wonderful stop along our trip water was very refreshing and looked very clean!

We saw the locals arrive and enjoy swimming for several hours.

I'm sure you can swim there but don't risk your relationship on my advice. :)