For your last note about the pedal to the floor, bleed air out of the brake system. But there's a good chance that it won't be a permanent fix; then you need a new master cylinder.

Handbrake warning light comes on for either of 2 reasons...
1 - Flaky switch at the handbrake lever. If the bracket is bent, the lever doesn't push the switch properly.
2 - Float-switch in the brake fluid reservoir. Low fluid or flaky switch.

That warning lamp circuit ties into the ABS. When the ABS sees that, it inactivates itself. But if it stays on, it stores an error code about handbrake being on.