Yes, but only if you get there EARLY, like by 9 am at the latest. Or you can take a (paid) trolley or Jeep tour and let them drive so you can enjoy the view. You can still drive thru later, but it's a bumper-to-bumper crawl.

Probably in the shady spots, not in the sunny ones!

Yes. However parking will be very limited.

No gates; open 24/7. Arrive early to beat the crowds.

We did not have a reservation. There is a guest center & gift shop. I'm sure donations are welcome!

I think you can drive almost all of the park. There are a few pull-over spots to see areas that would be difficult for handicap or walking challenged forks.

You should plan on 2-3 hours at Garden of the Gods. It has well-maintained trails and the scenery is beautiful. There is a lot to do in the area so I would recommend 2 - 4 days depending on your interests. And consider staying in Manitou Springs. It is really close to most points of interest and has a great downtown area to walk around.

yes go on a weekday

Most of them, yes

There is no train between Denver and Colorado Springs so you will need a car. You can do both in one day, but you have to watch out for the weather on the peak. If you are not experienced at mountain driving, you may want to reconsider doing the peak as it is a harrowing drive. You could still experience being in the mountains by driving up to Woodland Park and the Rampart Range without actually doing the peak highway. As to Garden of the Gods, take your time and walk through it. There is so much beauty to soak in.