I triggered Chandra, Fire of Kaladeshs ability, and as it's exiling ability twas triggering, an opponent played Swords to Plowshares. Would she stay exiled that way, or return to the field due to the latter half of her ability?

She would stay exiled. The trigger says "exile... then return" and the exile never happens because Chandra's already exiled by StP, so the ability sees nothing to return.

Relevant rule is 400.7: An object that moves from one zone to another becomes a new object with no memory of, or relation to, its previous existence. This rule has the following exceptions.

If you had a red instant in hand, you could cast it in response to StP, untap Chandra, tap to deal damage, and get another exile+return trigger to resolve first!

That makes sense. Thanks for the info!