Literally.

As I started off this morning, about 30 seconds into my ride I remembered that the front disc had been noisy enough on the last ride that I wondered if the wheel was in straight. So I came to a gradual stop on the 14% downhill that starts my ride. As I went to open the quick release, I accidentally touched the rotor with my thumb and it burned me. I mean, it burned me so quickly and intensely that I thought I had cut myself on something. It took a minute for me to realize what had actually happened.

I knew disc brakes got pretty hot, but I was shocked to see it heat up that much that quickly. Is this normal?

Imagine all the friction it would take to stop your bike if you were using your foot to stop. Plus they stopped them much quicker than your foot could. Ya brakes get hot.

Braking friction pretty much converts rolling energy into heat. The rotors on downhill bikes can literally turn blue from the heat - and they're usually larger than what you have. So hot rotors can be perfectly normal. But its only 'normal' after braking. Its also possible you are dragging the pads due to por adjustment - which will also heat things up.

My car had a slightly stuck parking brake. The rear wheel was so hot that I couldn't touch it after driving less than a mile.

Yep. Brakes get hot.

I hadn't even thought about the rotor size. My new bike has a 140mm rotor in front, and I start my day with a tasty descent. The stopping power has been plenty good with the 140's but now I'm wondering if I should put something bigger on there just for heat management. I'm not a small guy, and my morning descent winds up with me going about 35 mph approaching a pretty busy cross street, so I've got a lot of energy to disperse in a short time with rather nasty consequences in the event of brake failure.

As long as you're not braking hard repeatedly over a short time, it should be fine. There is a lot of air going over the rotor as you move, which should cool it off quickly.

It's normal and for normal riding your 140's are probably fine. If you are on extended descents that are really steep and you are on your brakes for miles you can experience brake fade due to rotors overheating. I've only had it happen to me once and that was on a mountain bike.

Seems 140 discs have been a recent weight shaving choice, usually on the rear of road bike sets.

MTB utility XC seems 180 F, , 160 R common..

The newer style rotors like Shimano Ice with aluminum spiders and aluminum sandwiched in the rotor dissipate heat better. I use both SS and AL-SS rotors on different wheelsets, and definitely notice the difference. Newer pads and larger rotors are a better at heat dissipation.

The real concern is not the rotor heating up to the touch, which is normal, but brake fade and boiling hydraulic fluid from the brakes overheating.

I remember years ago I had a car that had a front caliper stick on me. I was about 5 miles from home when it happened, so I just decided to give 'er a little more gas and force my way home. I made it home but when I got out of the car and looked at the wheel the rotor was glowing red like an oven burner. I'll never forget the smell either.

300-500 deg is acceptable for bike discs during race contexts. To an extent that is part of what makes them less succeptable to contaminants and water since it allows them to heat off. The heat by itself is not a really major concern other than when you rebuild the hubs(or have it done) make sure it is with disc grade grease as the inferior grease can melt out.

Talk about hot rotors