Can a hybrid drive be used as two separate drives? For example, is it possible to install the OS in the SSD portion and use the HDD for large media files? Will the SSD's performance be negatively impacted since both the SSD and HDD are sharing the same SATA cable (and thus less bandwidth for the SSD when writing/reading from both drives)?
If you are talking about the Seagate hybrid drives (and those are the only ones I know off which use that name at the time I originally wrote this), then No.
The drive just acts as a single regular harddisk. The firmware on the drive keeps track of which sectors get read a lot and moves those to the flash cache. It handles that completely independent of the OS and the OS only sees one normal HDD.
However there are also situations where a small SSD and a regular HDD are added to the system and software is used to tie them together. These can be configured either as two independent disks are as one disk with caching. To configure and use this you will need software/drivers. I have now heard these referred to as 'hybrid drives' even though that makes no sense to me. I would just call them 'two drives, one of which is a plain drive but very small and thus mostly useful for caching'. This might be due to packaging where the two drives are somehow delivered as one physical unit.