Ugh medical bills and dealing with medical billing people. I had to go through something similar recently, though it was on a much smaller scale.
(1) Possibly. I'd probably start by calling your insurance company. Try to understand what is different this time from last time, etc. Sometimes things are coded wrong, and you may need to call the hospital to get codes changed to lower the cost. Compile a big list of questions (and have all the codes handy) and just go line by line on the phone. If you don't have a line-by-line from the hospital with billing codes, get that first. Always document everything - date, person you talked to, what they said.
(2) Hmmmm...The hospitals may be able to run a quote for you, through the insurance. I had this done with a dentist on the cost of a procedure when they weren't sure how much it would cost me. This could take a bit of time, but sounds like it would be worth it. They might not even need to run a sample claim to give you a quote. You don't know if you don't ask! Like NCGal said, you could also see about an out of pocket quote.
(3) Quite possibly. Figure out why the cost was so different this year, and what you'd have to pay with the HDHP vs. a different one. Factor in tax savings with the HSA, etc.
(4) I'm thinking yes, though it might cost a few hundred more (I'm guessing anyway - you'll get a better idea once you start understanding each line item). She should have access to all medical files, including MRI scans, and at the very least she could request these files, then go to the other hospital. The hospitals would likely share the files, too, if requested.
Sorry you have to deal with this. :-/ If all else fails, you might want to contact a medical billing advocate - someone who regularly deals with this kind of stuff. That would be some fee, though, either up front or some portion of what they save you on your bill. I'm all for do-it-yourself, but in some cases, with the mental exhaustion something like this entails...might be worth it.
Good news is you now know better and will likely be able to reduce your costs in the future one way or another! This kind of stuff is always easier to deal with before the procedure or test has been completed.