[RPG] Does a stunned creature roll Initiative with disadvantage

Although contests are RAW, how to run a contests is explicitly made the DM's responsibility via their own good judgment of the specific situation at hand. That means that, RAW, as soon as a contest is on the table (so to speak), it's in the DM's hands to make any and all adjustments necessary for it to make sense.

In this way, the rules are — again, by RAW — not telling the DM how to do it, only giving them a basic tool to riff off as necessary. Some situations will call for using that judgement to give a stunned character the chance to oppose a grapple, while most situations will probably not.

It might have been nice to have corner cases like this nailed down instead of punted to the DM, but relying on DM judgement is intended to be a feature of 5e, so this is quite intentional. DMs are smarter about the situation in front of them than a rulebook could ever be, and 5e's design intent is to harness that constructively.

I would say that Alex can no longer finish their turn, as they are immediately Stunned and therefore incapacitated.

Stunned

• A stunned creature is incapacitated (see the condition), can’t move, and can speak only falteringly.

To me, this is no different than if the Monk had attacked normally, without using the Stunning effect, and reduced Alex to 0HP, therefore knocking them unconscious (which incidentally is also incapacitated), which would take effect immediately.