You don't need to take anything offline to run a backup. If you take the database offline, you can't run a backup in SSMS.

As far as the mapped drives, the server doesn't see drives you've mapped in your session. The service account for the server has it's own mapped drives.

What I would say here is run the full backup. use a UNC path. Then switch to simple mode. This will generate a checkpoint and the log chain is broken. You should be able to shrink the log then.