You should be able to verify the bad ring gear yourself by inspection underneath before starting the job. Yes removal of the transmission or unbolting and sliding it back is required to replace the part. Rather than replacing just the ring gear change the flexplate and gear as one.
A used part should do fine and with a little luck and no experience you could do the job on a saturday for example. This is not rocket science with the guidance available on site.
If the part used cost say 25.00 will you work a day for 725.00? Under a pit or hoist even the first time could go faster. Use new bolts on the flexplate to crankshaft though. You will need to buy, rent or borrow a torque wrench to do it properly. Getting a friend to help never hurts.