It shouldn’t. Was it the lower cup or the upper one? Neither should just pull out but the lower one would be of greater concern since it would suggest that the headtube is ovalized or damaged. Inspect the headtube for cracks and use a caliper to measure the diameter to see if it is consistently round.

Technically yes, but unless it wiggles, it will be fine.

How do the Bearing Races look, after cleaning? Is there ANY pitting/discoloration on the bearing Balls?
if both are clean and not pitted, then the "notchiness" was the Old Grease in lumps on the bearings/races, and all you need is to COMPLETELY clean the bearings and re-install everything, the Adjust them to the correct "No Slop, No Drag" setting.

Final Tightening of the lock ring, then the top nut WILL slightly tighten the Bearing adjustment, so a few tries will be needed. and sometimes, you'll get real lucky and everything works, firs try.

Tip: Double Check the install DIRECTION of the caged Bearings... they won't work if they aren't in the correct orientation to the races.

It would raise an eyebrow and prompt me for further investigation. I run a volunteer bike shop and work on hundreds of bikes a year. The only time that a bearing cup just came out on its own, there was a crack in the head tube. They are normally pressed into place.

I've seen it happen on TIG welded frames lacking reinforcement on the ends of the head tube (which isn't the case here). Abusive riding can cause the lower end of the head tube to flare and allow the cup to loosen. It's possible, but unlikely (IMO), to be the case for the OP's bike. It could just be a poorly prepped frame. In any case, a bearing retaining compound, like Loctite #609, can hold the cup securely.
https://www.amazon.com/Genuine-Locti.../dp/B07BCDYD2C

An adjustable wrench with a breaker bar. Use the front wheel to brace against. Very simple

I don’t like the secure the front wheel method, there is chance, if it is cemented, that the wrenching could affect the fork alignment. Imo much better off securing the fork crown or the handlebar.