How Do Bikes Stay Up?

The surprising thing is that it is way more than "a few feet" . Apologies for repeating myself, but watch [this video] (And that is not on smooth terrain) And nitpick: typical bicycles aren't symmetric left-right. The chain and gears make them heavier at the right. Most mass is close to the center line, though, so its moment is negligible, especially when ridden by a human.

Bicycles are surprisingly stable as long as they are able to maintain some momentum. minutephysics had a great video about this too: How Do Bikes Stay Up?: [this video]

As far as bikes go, they're designed to stay upright by automatically turning the front wheel when it is tipped in any direction. [this video]

Objectively not true. Here’s a video of a unmanned bike rolling over a curb without any trouble: [this video] Stuff on the road will only knock the bike over if it stops the motion of the bike first. The need for motive power was already acknowledged.

I'm not sure how closely that applies. We didn't, but mostly because people hadn't really studied it and hadn't worked through the mechanics. As soon as we really started looking AFAIK the solution came pretty quickly; it's a combination of 3 things that all make the bike steer towards where it leans: gyroscopic precession, head angle and CoM of the front steering section. All 3 of those things can be tuned to be larger or smaller and that affects the velocity required to achieve stability. [this video]