I know I can use a 3Gb/s SATA hard drive with a 6Gb/s SATA port, but can I use a 6Gb/s SATA hard drive with a 3Gb/s SATA port? I am unable to find a definite answer.
Yes, they are backwards compatible, in the same way a USB3 hard drive can work with a USB2 port.
There is a slight chance that you'll need a BIOS update, though. My Gigabyte MA785GMT-UD2H board, which is SATA II, would not see an OCZ Vertex 4 128GB drive, which is SATA III, in AHCI mode. IDE mode was fine. Gigabyte sent me a BIOS update that fixed the issue. I read of 1 other Gigabyte customer with the same issue and same fix, but for a different board. The board had the AMD 785G chipset like mine, though. I don't know if it's a Gigabyte+785G issue, or a 785G issue. Jan 4, 2013 at 14:00
Found the link about the other board: posting. It seems Gigabyte doesn't post these BIOSes online. You have to get them through support. Jan 4, 2013 at 14:13
Actually Wikipedia says that not all SATA chipsets support backward compatibility. In general, it should be safe, but it is impossible to be sure without knowing chipset model and possibly also hard disk model.
Yes, you can. SATA III is backwards compatible with SATA II, which is backwards compatible with SATA I
You'll be able to use it yes, but you most likely won't get some of the SATA III protocol benefits. For normal desktop usage this probably won't be an issue.