Guessing you replaced the battery due to it being bad. Are you sure you're not dealing with a charging problem ? The only function of the batttery is to start the vehicle. When the vehicle is running anything to do with electrical in the vehicle works on the alternator. I would suggest having the charging system checked on the vehicle cause the voltage regulator which is internal in the alternator may not being working all that well when it has a load on it. Just a thought but that's where I would start,,,,
Check for a frozen cable by pulling underneath cables that lead to the rear brakes. Could be a cable, or pivot. Sometimes spraying the cable or pivot with penetrating oil can free it up.
Its a very easy fix, you will have no problem doing it yourself. The window most likely simply came out of its track and therefore cannot be engaged properly. Uness there was some damage to the vehichle, I doubt that anything is actually broken.
The fix does involve removing some interior trim pieces and then adjusting the glass back on track. I have included a detailed set of diagrams and instructions which explain exactly how to fix it.
I’m happy to assist further over the phone at https://www.6ya.com/expert/jeremy_cd01d4e4178e5a11
