Can you Temporal Breach The Ball and play it again?

Yes — as long as The Ball is currently a unit gear (via Lady of Clockwork), since Temporal Breach only banishes "a unit" and The Ball is printed as a Gear.

The banish works and it gets replayed to that same location for free, ignoring its 5-energy cost (the spell specifies the location, which is the carve-out to the "Gear only play to Base" rule, Rule 149.2; Deploy also permits it at a battlefield, Rule 830.1.b.1). It comes back as a plain Gear, not a unit gear — changing zones makes it a new object and wipes the Legend's type change (Rule 124.1), so you'd have to re-choose it at the start of a later Main Phase. Good news: because it has Deploy, it's not recalled to Base (Rule 830.1.b.3), so it just sits at that battlefield ready.

And yes, you get the trigger again — replaying it is "playing" it, so "When you play or move this" fires: draw 1, or deal 2 to enemy units there. Opponents get a window to respond to that trigger on the chain. If the Breach came out of Hidden, the target had to be at the battlefield it was hidden at.