Backing up Oracle home directory while instances are running

The very first thing I'd try is copying the surviving control file then restarting the DB - the loss of the controlfile is what caused the DB to crash, and if the redo log group happened to be INACTIVE at the time, then I'm pretty much OK. I'd find this out from the alert log. So long as there is a disk in the right place, I can issue CLEAR LOGFILE and they will just be recreated.

If not then we have a problem - the SCN in the controlfile is ahead of the DBFs and we've no way to get there. We have no choice but to perform a point-in-time-recovery, and we have to discard those DBFs as well. In RMAN, restore database (assuming controlfile autobackup), recover until the highest SCN in the archived redo logs, then OPEN RESETLOGS. Since I have the cumulative incremental, I don't need the differential incremental (I'm not sure why you'd mix the two types? Also why if we had several disks for redo why we'd put an entire group on one disk rather than multiplexing all groups).

(There is probably a much better answer involving flashback, but I have to confess that I have only used explicitly created flash restore points so far, never run through a DR scenario).

The REDO logs should be copied.

Shutdown doesn't trigger a logfile switch.

If you are changing the paths for the files, then using the "create controlfile reuse noresetlogs" method is the best. If you do move it without REDO logs, then archivelog current and then use the "create controlfile set resetlogs" method. In general though, moving the redo logs are best.

--EDIT--

Based on what you said, you are mounting these in the exact same path, then do your self a favor and just put all of the files, DATAFILES TEMPFILES CONTROLFILES REDOLOGS, etc in the exact same file path. Nothing else to do and it comes up fine.

-Richard