No, you do not have to trek at Badrinath. The route is motorable so trekking part is not there. However there is a foot bridge that one has to cross to reach the temple and to avoid congestion, the vehicles leave you 300 meters before the bridge.

It completely depends on your perspective. Thousands of pilgrims trek to Kedarnath Dham every year and they enjoy it. At the same time, there are others who book their shuttles and fly directly to the temple. Mostly people who are scared of trekking will find Kedarnath trek dangerous because they are not used to the habit. However trekkers who live for such things find it the best track for trekking. During the monsoon season, the track becomes slippery and then it is risky to climb and that is one of the reasons that Kedarnath and other Dhams remain closed during the monsoon season.

Yamunotri Temple is not motorable. The last point at which your vehicle drops you is Janaki Chatti from where one has to trek for 6 kms to reach the temple. Yamuotri Trek takes 2-3 hours to complete the trek. You can find palki/pony over there and you can hire that as well. If you do not wish to trek, you always have an option for a pony, however if you want adventure then trekking to Yamunotri Temple is the best choice for you.

Compared to Kedarnath Trek, the Yamunotri trek is less difficult. Most of the trail is concrete, however there are some steeps that one should watch out for. The path is approximately 8 ft wide, so it gives plenty of space for pilgrims to walk, but the horses and the pilgrims use the same trail, so one should be careful.