1. Sounds as though the ring was either picked up by the cleaning crew and turned over to UA or to the RIC L&F. I would call L&F again and get a definitive answer. The fact that items at gates are "usually" turned over to the carrier is not definitive. A ring such as you describe would stand out. This is particularly true because the ring could have been picked up by a good samaritan who turned it in elsewhere in the airport. Finally, it may have been picked up by someone who put it in their pocket, If that is the case, this is all fruitless.
2. UA, like most carriers, works through an online form. It is available on its website:
https://www.chargerback.com/ReportLo...ustomerid=2474
If you have not yet filled it out, please do so immediately. It is the fastest, best (and only) way to deal with this short of going to the airport (convenient or not).