So there's this 500GB HDD with 8MB Cache and it's interface is SATA II (3GB/s). Will it be fine for gaming? What about use in a SATA III (6GB/s) Port?
It will run at SATA 2 speeds if it's really a SATA 2-only drive.
Whether it's "fine" for gaming depends on your definition of "fine", and what games we're talking about. Many games only hit storage to load the game and not thereafter, so the only impact is initial load speed. If you're willing to tolerate that then OK. Other games, typically open-world type games with lots of storage based scene data, hit storage during gameplay and then an HDD can be seriously annoying.
By the way, the real life difference between SATA 2 and SATA 3 is minimal for things like gaming. A game load might go from 15 seconds down to 12 or 13, but probably not less.
sata speed and cache are nonsense for gaming?
"for gaming" isn't a definitive phrase. If you re-read my comment, I tried to distinguish between two general types of games, one that only hits storage when starting up, and one that hits storage during gameplay. The second type of game is much more sensitive to storage speed, and I wouldn't want anything less than an SSD. (All SSD's are so much faster than any hard drive that it's not worth discussing the minutiae of hard drive speeds.) The first type can run on punched card storage if you're willing to accept the start-up delay, it's all about what you are willing to tolerate.