I know this isn't the answer you want to hear, but I don't think priming and painting will hide the tape.
Before you apply any more primer, I'd add another thicker coat of mud. Slather on the mud thick first - it doesn't have to be smooth. Then smooth it out by angling your 12" knife so it's almost parallel with the drywall as you drag it along the surface. You'll have to press fairly firmly in order to avoid air gaps appearing in the thick coat you're smoothing out.
You may have to get a feel for it. If you push too hard and can see the tape again, redo that section until you get it just right.
Wait for the thicker coat to dry, then sand the edges as smooth as you can get them. Very carefully/lightly sand over area covering the tape if you didn't get it quite smooth. You may need to do a second coat along both sides of the thicker coat to make it join the drywall surface smoothly.
You raise a number of issues.
How to fill an irregularly shaped hole?
To cut drywall (or any panel) to an exact shape with non parallel sides is hard. So what we usually do is make the hole regular. You can trim away some of the remaining drywall, preferably so the edge runs along the middle of a stud, parallel to another stud or the corner. Then cut a rectangular patch to fill the hole, leaving about 1/8 inch gap.
How to make a patch sit in the same plane as the wall?
In general, we want the whole wall to be on the same plane (when you place a straightedge on the face of one section, it sits flat on the adjacent section, neither gapped or raised). If a patch will sit below the level of adjacent wall, you can put shims behind the patch just thick enough to raise the surface to flush. You can use trim boards or tapered shims to get the right thickness, tack them in place with brads, and drill through them when installing the drywall. If the patch sits above the surface, its usually best to use thinner material (but don't violate fire laws about minimum thickness).
In your second picture, the patch looks like it is about 1/2+ lower than the wall. You should shim it out. Even though you could use tape or drywall corners to cover this zigzag, it's a weak, thin joint and will look odd.
How big a gap is acceptable?
Between panels, 1/4 inch is fine. Taping compound and tape will cover it with little difficulty. Putting on a coat of compound before setting the tape will fill the gap. If you are using mesh self adhesive tape, forcing a bit of compound through the tape into the gap couldn't hurt.
If you have a gap that is approaching 1/2 inch or more, you can run into a problem with shrinkage and strength in the joint. Consider recutting the patch. You could use a thin sliver of drywall to partially fill the gap before taping, but it will be a bit more of a challenge to get it level and smooth.
Best Answer
I doubt that a 4x4 hole that's fixed with mesh tape would be durable over the long haul (my limit is about an inch), but if it was somewhere that never gets any bumps or touching, then maybe it would be fine. (You're replacing something that's 1/2" thick with something that's much thinner...)
On a practical level, you're apt to have problems with the mesh sagging/bulging. If you're brave enough to try this repair, I'd suggest you use a stiff mix of setting compound both to butter the edges of the hole and to do the first coat. Try and press as much through the mesh as possible. (I know that "stiff mix" and "press through" are somewhat incompatible.) You might also look into the 6" wide rolls of mesh tape. They'd certainly have more structure than 3 strips of normal mesh.