[RPG] Can a character shove an enethe who is already prone

Yes

"If the attack involves other damage dice" sounds like a pretty solid indicator that "Superiority Dice" would be multiplied, if a maneuver were used that adds the result as damage.

While not applicable to everybody, that sounds like an awesome way to add extra damage to multiply (not that superiority dice are really game-breaking for additional damage) after learning that an attack hit... but I'd allow it, regardless.

You can add it after the roll, but not after someone else uses a reaction to respond to the result of the roll (i.e., because the roll indicated a hit and that result was declared):

You can use this maneuver before or after making the attack roll, but before any effects of the attack are applied.

The attack's success causing someone else to choose to expend a spell slot and their reaction is definitely an “effect of the attack” being applied.

It sounds though like this might have happened when another player simply saw your die roll, rather than after a final roll result was declared.

If your opponent is casting shield because they directly observed your die roll and didn't wait for you to declare the final result, that's a bit messy but it's entirely possible to untangle it, and fairly to everyone involved: say you weren't done, apply the superiority die, and then the opponent can choose to (or not) react with shield.

Of course, if they react with shield after you've declared that your final roll is a specific number and it hits, then you've committed yourself and that kind of “rolling back” the action declarations and resolutions would likely only happen if everyone was feeling generous.