why do I have huge log files (SQLFT000060005.LOG.x, 3+Gb) with recent timestamps when I haven't used SSMS for 4 months?
Log files are created by the SQL Server service, not by SSMS. If you installed with the default settings, you can stop and restart the service multiple times (or just call sp_cycle_errorlog repeatedly), and that should cycle away the big log files that are there. They'll come back, though, over time - even if you don't open SSMS.

If you don't need SQL Server to be running anymore, disable the service or, if you don't need it at all, uninstall it. You can also uninstall full-text search if you aren't using it (or never needed it). Those FT logs might be from that service.