I'm trying to restore a database backup made in SQL Server 2008 R2 to the 2014 Express version. Is this even possible? If I choose restore from device and select my backup file, I see no database in the drop down.
Yes, you can, as long as the database is < 10 GB and doesn't use any features not supported by Express (you didn't mention what edition was being used in 2008 R2).
Stop using the broken point-and-click UI, and instead use a proper
RESTORE DATABASE command:
RESTORE DATABASE DBFrom2008 FROM DISK = 'C:\backups\2008backup.bak' WITH REPLACE, RECOVERY, MOVE 'data_file_name' TO 'C:\...\database_name.mdf', MOVE 'log_file_name' TO 'C:\...\database_name.ldf';
If you get errors from that (after replacing all of the obvious bits that are specific to your database and system), post those.
Depends on what edition. What you're doing is basically a side-by-side upgrade. See full list of supported upgrade paths at https://technet.microsoft.com/en-us/library/ms143393(v=sql.120).aspx