I don't have a degree in anything (including CS, which is probably why as a self-taught programmer I reached a ceiling - but that's another conversation).

When I worked at Uber, a CS degree was a soft-prerequisite for all Product Mangers (they hired me, they hired a few other people without CS but it was rare). I hear Google is the same. At Uber I also saw a couple of Sr Engineers switch to PM.

In terms of suggestions, I would focus on where you already work and see if they have interest in helping you switch to the PM track. There's lots of ways an engineer can straddle the two and begin to do some PM tasks.

I don't know if there's any difference in mobility between PMs and Eng Managers. I would say that generally there are fewer PM roles in an organization than Engineers so that is something to consider, but good PMs are always highly sought after.

I think this is highly dependent on the company. I've worked at a few of the top few tech cos as both an engineer and a product manager, and I don't recall working closely with anyone who came from consulting, although there are probably a few. A handful have MBAs, some come from engineering, some have weird liberal arts degrees. It's a very diverse group of people in my experience. It can be a good career path for engineers who are more excited about the problems they're solving and the people they're solving them for than about the raw technical challenge.

Upward mobility is also highly dependent on the company. I've found some companies to be more product-driven, and some to be more engineering-driven (e.g. goog). Upward mobility for product managers is obviously better in product-driven organizations.