It sounds like you have your hands full so let’s look at your surging RPM. Normally, RPM surges can be caused by several things. Its possible that the torque converter has one or more broken vanes in its impeller. This would cause a weird loading pattern for your transmission fluid and could result in surging. Another possibility is that the clutches are slipping in a major way, causing the engine to overrev as the transmission hunts for the right gear. And, it is always possible that the input shaft is damaged and slightly out of round so the input it is delivering to the transmission is oddly loaded, causing the surging.
As you can see, there’s a whole range of problems that might be possible and which exhibit surging and overreving.
Here’s what you absolutely have to do to find the cause: get to a good transmission shop — or even the dealer — and have the facility put an OBD-II scanner on the diagnostic connector under the dash. Watch the readout and get a hard-copy of the results. I think the chances are good that you will see error codes for a failing transmission control module or the module and sensors. It is also possible that the solenoids are shot and contributing to the problem.
Honestly, I think you are facing the need for a transmission replacement. That will cost you about $3,300 — more or less, depending on the location. There are so many potential problems in your transmission that it’s the only way to handle things, even with the diagnostic printout.
As to the other problems, it sounds like you have some metal-on-metal issues in the engine itself. If that is the case, then, you could be facing some really expensive problems as it will cost you thousands to replace the engine.
Frankly, maybe it is time to consider a new conveyance. After all, your car is 13 years old and it is likely paid to the ground. I’d take whatever equity I could get for it and buy a relatively recent used model. The reason I suggest used is that the former owner has already rung the engine and transmission out so you will likely gain some years of pain-free driving. And, yes, your bills will go up monthly, unless you pay in cash. However, with leases today averaging much less than $200 it only makes sense to get some more modern wheels. Let me know your decision.