[RPG] Can a Battle Master fighter continue to use maneuvers after they have spent all their superiority dice

Yes, a Battle Master can give up their single-attack Attack action to use Commander's Strike.

You appear to be confusing the Attack action with an attack.

The Attack action is one of the things you can choose to do with your action; others include the Dash action, the Cast a Spell action, and the Use an Object action.

An attack is anything you do that causes you to roll a die to overcome someone's AC. Trying to hit with a weapon is an attack, but so is using a spell that rolls against AC, such as eldritch blast. You can make attacks without taking the Attack action, e.g. attacking with a spell using the Cast a Spell action, attacking with a magic item using its own unique action, or attack with a weapon using a reaction (e.g. using the Ready action, or making an opportunity attack) or bonus action (e.g. from Two-Weapon Fighting).

The prerequisites for the Battle Master fighter's Commander's Strike maneuver are that you:

• take the Attack action, which always allows at least one attack, and
• forego one of those attacks, even if it is the only one

The rules for grappling and shoving a creature are similar.

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.