1. Before doing
anything major, make a
full TWRP backup and save it to your PC. (That includes updates on phone that are still getting updates.)
2. If the ROM isn't an .md5 file (or a .tar) file, get one - even the stock ROM. Keep it on the PC. Put the phone in download mode (Home+Volume Down+power until you see the warning, then Volume Up to go to Download, if I recall - it tells you on the screen). Run Odin. Load the ROM file into Odin. Here - follow the writeup I did - it includes everything (except that you might not fine a Note 3 ROM at Sammobile any more) -
[Samsung] How to flash Stock ROM via ODIN.
Once you're back to stock, you can flash whatever other ROM you were using by whatever method the developer set up - Odin, TWRP, there are a few ways.
And no, you don't do any resets when you replace the battery. It's going to go, and you just replace it. (Even the "new" one you got - which was made in 2013-2014 - is old and it's not going to last as long as the original one did.)