Kitchen rewire gone wrong
Yeah, the problem is your work methods. When you hit a problem, you're not stopping to iterate on the problem until it's fixed, as you should.  You are ignoring it and dashing off to do yet other work.   So now you have a set of compound problems that are all interacting with each other.  

You seem to think that one thing breaking breaks another thing, for instance you are now keen to replace the breaker.  The breaker is just doing its job, you have a short in the wiring.  If you don't know what to do, don't do random other things; focus on learning what you need. 

Given the depths of this mess, it may be beyond our help. 

Fortunately, it doesn't matter: it's all coming out. 

When you have the walls exposed, you must modernize to Code

It's not enough to swap the old wires for new wires.  You need to rearrange circuits and outlets to conform to modern Code.

In the kitchen, that is a big job. It involves quite a number of new circuits.  Some of them must be dedicated to the kitchen, so for instance it will no longer be possible to have a circuit that serves kitchen receptacles and also other locations.  

How that's done is beyond the scope of this question, but you have a lot to do.  It will entirely moot the problems you're having now. The upside is you get to do it all as new work, so you can start from scratch.  This also means you'll only need at most one GFCI per circuit, so that'll save you some money. 

Oh, but I can only do half the job

Then you have to do the half that you can do; that is to say, use the access you have today to prepare for when you are able to do the rest of the work. 

What you cannot do is use the excuse "well, we don't have ALL the access we need" to simply do nothing at all, and leave the old work wholly non-compliant. 

Also, the claim that "we can't get wires through that other part we're not remodeling" isn't really true at all. A competent old-work experienced electrician can get wires through all sorts of places you might not think.  Or you could simply expand the scope of the drywall work you'll be having done anyway, to gain a few key access points.