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.
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.