Can the Intel C600-series SAS controller RAID SATA drives

SAS uses higher signalling voltages than SATA so mixing those on the same backplane isn't the best of ideas, although you can get SATA interposer cards to allow SATA to be used on the same backplane.

Near line SAS however uses a proper SAS interface so there should be no issue with your intended setup even if you put them all on the SAS 5iR.

I would suggest using the SAS 5iR rather than the onboard SATA for a few reasons - the 5iR will be connected to your hotswap backplane and the onboard SATA won't. Also from my experience the onboard SATA on most poweredge's is just a basic SATA controller that doesn't support RAID in any way shape or form.

I don't have another server to plug them into, and even then connecting the drive to a different controller (while keeping the data) is not a certain thing.

Other then buying a RAID controller and plugging it into a workstation (in which case I still may not be able to connect the drive while leaving the data intact as this is a RAID controller, not a simple HBA), what are my options?

In most cases you can just get an identical or similar RAID card, plug in the drives and access the data. This will just work, and this is something you can test. And if you are concerned about 15 minute delays then make sure you have such a card as a spare.

Does something like a simple (non raid) PCIe HBA card that just lets you plug in an SAS drive exist? or better yet some sort of external enclosure? I can't find either on google.

Yes, ofcourse that exist. They are often the same pysical cards as SAS RAID cards, but with a different firmware.

Did you consider keeping a copy of all the data on the network. Sort of a RAID 1 (mirror) where one part of the RAID is a local drive(s) and one part is on the network?