1. Not sure if I'd say "slower" but crappy, nonetheless.
2. "huge" not so much, but definitely noticeable. DVI and HDMI provide a much clearer digital signal.
3. No, that really isn't a good idea. The adapters for it are pricey, and no, would not provide HDMI quality.
4. I don't know if it would "cause lag" or slow response times. However yes, VGA and HDMI compared to DVI have a rather small bandwith; not small enough for a normal conumser to notice, but still, DVI (or even Display Port) is always the better option.
Is your monitor VGA only? If it is, then it's probably old enough to have a bad resolution; that 760 might be overkill.
If your monitor is 60hz, then any fps 60+ will feel like 60fps. And you wouldn't notice any difference between 60Hz, 75Hz, and 80Hz. If it were 120+Hz you would notice.
And yes, there is a relation between refresh rate and fps, 60Hz is bare minimum for gaming. Your monitor is only capable of showing (this might be inaccurate, but for argument i'll use it to show a difference) about 1 frame per refresh rate cycle. This is why 120/144Hz are usually considered to be the best "Gaming" monitors. More frames+high refresh rate= smooth gameplay.
However, with high refresh rates you sacrifice color accuracy.
No, there'd be no point in upgrading from 60Hz--->75Hz. However, about overclocking, I did it myself, I got to 74Hz on my LG IPS display before the HDMI cable ran out of bandwidth. Overclocking anything is potentially dangerous for the part, so be careful. And don't do it with VGA.
Also you mentioned you have a 760, which can max out any game in 1080p, so i'm not sure why you'd go with 1366x768. However, to answer your question, no there are no monitors that support that res AND the refresh rate.
That applies to 3D, where a 120 Hz monitor renders 60 Hz per "eye." Aside from that special case, a monitor that redraws its image 60 times in a second syncs up perfectly to a game running at 60 frames per second.