Powering Arduino from 12v 1A linear regulator?

You're throwing away 12V of power (converting it to heat).

Assuming a couple of hundred milliamps of current: 12 * 0.200 = 2.4 Watts of heat.

Yep, that'll get hot.

It might work with a heatsink but you'll be back to square one if you add some more LEDs.

Connecting a few of the regulators in parallel allows a few more outputs to work before they all cut out.

Connecting regulators in parallel is bad. Don't do it.