Does it make a wrestler's moves and/or finishers do more damage?

Does it make the opponent's moves and/or finishers do less damage?

Does it increase the chances of a successful pinfall or submission attempt?

Does it decrease the chances of a successful pinfall or submission against the wrestler in question?

Does it increase the chances of a wrestler gaining and/or keeping the initiative in a match?

Does it work in some other way that I haven't thought of?

All you mentioned actually takes place.

But - this might throw you off - the wrestler's absolute push isn't what is used. It's the ratio of the wrestlers' pushes that is used to calculate multipliers etc.

For instance, if you have wrestler A with a push of 100 and wrestler B with a push of 75, A has a 100/75 ratio over B (133%) while wrestler B has a ratio of 75/100 (75%).

That doesn't automatically mean that if wrestler A does the move X to wrestler B, his move will do 33 percent more damage than usual - nor does it mean that if it's the other way round, 25 percent of the damage are deducted.

We're talking about potentials here. If a move has a damage rating of 100, with wrestler A it has a maximum damage potential of 133 while with wrestler B it has a maximum damage potential of 75.

In a nutshell, it's still possible for one particular instance of this move done by wrestler A will generate a damage of 30 while the same move by wrestler B MAY cause a damage of 70, however it's not very likely to happen.

Push ratios affect everything from countering to pinning, making submit, getting countouts (albeit pushes don't play a major role here) and even getting the nod if a match is tight.