I can't see why Find My Mac would make a difference, but turn it off and see what happens. Troubleshooting software can be a long process at times.
Since it just quit working, if you have a backup (hopefully you do) try restoring the application and its related files, which you found when you used AppCleaner. Restoring the application may say you can't do that. If it balks there might be another solution.
Answer to your second post, in theory no, but I would do a complete hard drive backup in Time Machine or use a cloner before reinstalling, just in case of an oops.
If you are not familiar with cloners, here are the two most recommended. I haven't used them:
http://www.shirt-pocket.com
http://www.bombich.com