When empowering Kharox and my opponent burns 3 can I select a unit from the 3 burned cards or does it have to be pre-existing in their trash?

Yes. You can choose a unit from the 3 burned cards.

Kharox's "Then you may do this:" creates a reflexive trigger (Rules 387.1), so "Choose a unit in their trash" is not a target locked in when the Empower trigger goes on the Chain. Instead, the burn resolves first, the reflexive trigger then finalizes, and you pick the unit from their trash at that point — after the burned cards have already entered the trash (Rule 355.5.b). The trash is unordered (Rule 108.2.c), so all cards there, including freshly burned ones, are valid choices.

This is different from cards where the trash target is chosen upfront (e.g., Spectral Matron's play effect), which lock the target before prior instructions resolve.