If you've got the 806, then chances are, the G10 of the 710 will be about the same. On mine, I find that to be sufficiently grippy. The handle does have a shape that feels like it could lead to slipping in some circumstances, though at the same time, I've always found that the handle is long enough that slipping onto the blade doesn't actually occur. The 710 is a great knife in a great size, great for gloved hands, even, but the lack of a good finger guard can feel kind of worrying at times, even if nothing bad ever ends up happening.
As for the Grip, the answer is yes, it will definitely provide a more secure grip against slipping onto the blade. For my hand, the ergonomics of the Grip are superior to those of the 710, though the 710 has better blade geometry, and just has the feel of a harder use knife.
But if you're looking for an ambidextrous folder with a good guard to prevent slipping onto the blade, look no further than the Rukus. The finger guard on that knife is probably the best I've ever seen, and the knife is twice as much of a tank as the 710 will ever be. As for whether the Mini Rukus or full size is better for you, that's for you to decide. The 615 is a relatively large knife, near the size of the 710. The 710 is huge. Both models have plenty of finger guard, and feel more secure (in my opinion) in the hand than the 710. Probably more so than the Grip, as well, for that matter.
And no, lint will not affect the Axis lock in any kind of negative way. Mine get pretty linty and I've never had issues with lock stability. I do not believe it would be an easy thing to make the Axis lock any less secure. Heck, the system doesn't even get any less smooth with excessive lint in it.
Bottom line: check out the Rukus. Between it, the Grip, and the 710, buy the one that appeals to you the most (from your description, though, I'd lean towards the Rukus).