What type of "bumper-to-bumper" warranty is this? Assuming it's a used car that is outside of the manufacturer's warranty, you're going to have a hard time getting the warranty company to pay out for anything. By all means try and fight them to get as much money as you can, but most of those warranties aren't worth the paper they're printed on.

As far as getting anything reimbursed for rental cars, there's probably nothing you can do there. Shops aren't required to give you loaner vehicles. It's a nice perk when they do, but you aren't owed anything if they don't give you one.

Assuming the goal here is to get the car fixed - there isn't much you can do aside from taking the car to a different shop that is capable of fixing it in a reasonable amount of time. Clearly this shop isn't interested in working on the car. Continuing to wait on this shop to get their shit together is probably going to be a waste of time that results in more financial hardship as they misdiagnose things and require you to continue paying for a rental vehicle.