Yes, even economy award tickets are refundable, so if you buy a ticket with points, you can later cancel it and get the points back.
Yes, you have time to meet the sign up bonus on some credit cards before your trip.
Award booking becomes easier and easier the more flexible you are. Most important questions: How many people are flying with you? Is it possible to fly on different planes? (E.g. if it's you, a spouse, and three kids: probably not; if it's you, your spouse, and your siblings-in-law, then it's probably fine) Second, what are your date ranges? Do you need to be there on a specific day (e.g. wedding?) How long are you trying to stay?
Look at other major and medium airports near your destination. In this case definitely look at Rome-FCO, which is a half hour from Rome city center by train, and under 2 hours from Naples. Or 2:30 if you drive. (Though Napoli drivers are considered horrendous!)
Are you asking about United and American for a specific reason? Do you already have a pile of points with one of them? Or are other airlines possible too? (TBH it's likely these are indeed the best options in your situation.)
The most informative resource I've found are the subreddits r/churning and r/awardtravel. You can read up there, however they are pretty mean to newcomers so I don't recommend posting there, just reading. Mustachians are also knowledgeable, and much kinder.
If you answer these questions, I am eager to give more specific advice!