How to tape/support drywall seem next to cabinet

I'd prefill any gaps and then flat tape the drywall up to the cabinet. Once the drywall is finished - caulk the drywall to the cabinet.

I'll be adding a backsplash. Should I be concerned about flex at the top/bottom where the sheetrock meets the bottom near the counter and the base of the cabinets?

Essentially I'm getting conflicting feedback on the grounds wether I should add a nailer at the seem. This dude did _not_ https://youtu.be/YZBqYskcs2w?t=458 however someone over at reddit is saying I should add a vertical nailer at the drywall seems. (top and bottom)

If you are going to tile the backsplash you can skip the flat tape.
It's always best to have a nailer or some sort of backer at the seam - especially if the drywall is expected to give.

I don't think the tape provides much in the way of strength; if you're worried about the drywall moving from the weight of tile, put in some cribbing across just below the cabinets, flush with the studs, and screw the drywall to it. Otherwise, I don't see the need for tape there, as it will be covered with tile and caulk.

- Did this dude put drywall pieces where CBU is supposed to go?
- Did this dude put a moisture barrier?
If he did, I missed that part.

To your 2x4 horizontal pieces question, I would put them in the red zones.

A backsplash isn't considered a wet area so installing the tile directly over tile is fine.