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.
I would say you have too much compound. You should be able to press very firmly and pretty much squeeze out the excess compound. The thin layer that remains will provide good adherence of the tape. You want the compound to fully coat the back of the tape, but no more.
Best Answer
You'll be fine, though you might want to go buy a bag of setting type (dry powder, mix with water - sets, rather than drying out like the stuff in a bucket) joint compound, both because it's the right type for use with mesh tape, and becasue of this tid-bit from USG:
From this document: http://www.usg.com/content/dam/USG_Marketing_Communications/canada/product_promotional_materials/finished_assets/cgc-construction-handbook-ch5-finishing-drywall-systems-can-en-pdf.pdf
As the real 'rockers say, mud covers a lot of sins...or gore.