When Ziggs, Explosive decks pile lots of bombs on one battlefield, and they are detonated (let's say 4 bombs), does every single bomb have to pay deflect cost? All of my online opponents, especially Irelias, are trying to make me pay the deflect for every single bomb (so 4 bombs = 4 recycled for deflect 1).

Yes — each Bomb is its own object with its own death trigger, so 4 Bombs detonating means 4 separate triggered abilities, and every one of them that chooses a unit with Deflect carries its own Deflect cost (Rule 809.1.c: the cost is paid "for each time they choose me"). If all four pick the Irelia, that's Deflect ×4 — your opponents have that part right.

But they're leaving out your out: Deflect is a mandatory cost for spells, while for triggered abilities you choose targets, then at finalization you may simply decline to pay, and that trigger is removed from the chain with no effect at all (Rule 405.2; FAQ #10636). So you pay for as many or as few of the four as you want — no one can force you to pay for all of them. Two things to keep straight: declining kills the whole trigger, so if the Bomb you decline was the one whose trigger would "kill your other Bombs there," that chain reaction stops too; and if there's a non-Deflect enemy unit at that battlefield, choosing it costs you nothing. Also, Deflect power can be paid with any domain (Rule 809.1.c.1), so don't let anyone insist it must match something.