I swear, when i tried this earlier today, it hadn't worked ("my attempts failed"). Obviously, i must have done something wrong. I am wondering why this doesn't work:

Cases[expr, Equal[_, _], Infinity]

while this works:

Cases[expr, Equal[_, a_], Infinity]

But yes, that's exactly it, thank you very much @Henrik !

btw not very interesting, just trying to pick up some basic Stochastik knowledge and figuring out how to make Mathematica do all the work haha. Today, for example, i learned that game's pretty much over fast, as soon as the random variables $X$ and $Y$ are dependent. Or, when $X$ is something non-trivial, M. can't be helpful either because it doesn't know the distribution ... which is the thing that you're looking for in the first place lol. So one must calculate the distribution "manually" with combinatorics, programming, tree diagram, or random numbers simulation. Just in the case, when you know the joint distribution beforehand and feed it as input, then M. becomes an unstoppable power house. Anyway, i am still at the beginning of all of it. I barely understand the difference between EmpiricalDistribution and HistogramDistribution, two very similar functions, so confusing :D

To prevent early evaluation

Cases[expr, HoldPattern[Equal[_, _]], Infinity]
(* {y == 2, y == 12, y == 3, y == 11, y == 4, y == 10, y == 5, y == 9, 
 y == 6, y == 8, y == 7} *)