1.) I think the tiger would win.. I don't mean to imply that grizzlies are weak or incapable or anything, but I imagine that their fighting ability pales next to that of the tiger. Tigers seem so much more agile and.. I dunno how accurate this is, but they just seem more fierce, like they'd be more of a threat when backed into a corner than the grizzly. I picture tigers really hunting their prey, too -- stalking and attacking animals like wild boar, buffalo and even bears while grizzlies primarily eat herbivores (as far as their mammalian prey goes). So tigers just seem more capable in a fight.
2.) Hippo. I think that an aggressive hippo would just plow right into a polar bear.
3.) Gorilla. I mean, gorillas are some scary sh*t! But while I know for a fact that they are immensely powerful animals, I think my answer is based on the "Tarzan" movie.. I mean, come on, Kerchak totally could've taken on Simba.. But yeah, without the ability to stalk and ambush the gorilla (since they're pitted in a cage), I think that the gorilla's brute strength would win over the lion's agility and biting power.
4.) The jaguar has developed an incredibly strong bite force as well as a unique method of killing by piercing a certain part of the prey's skull. Because of that, I think the jaguar might have the advantage over the Sumatran tiger.
5.) I think that without the ability to ambush the black bear, and because the normally shy black bear would be driven to defend itself with everything it has, I guess the mountain lion would lose.
6.) For the last questions, I assumed that all of the animals in these hypothetical matches were at their physical prime and had primal motivation to fight for their lives. This question is harder, though, because while the modern Mastiff is a formidable animal, their fighting days are long gone. Long ago they were frequently used in lion-baiting, bear-baiting, dog-fighting and even against gladiators. They've since been bred to be pretty mellow, and while there are Mastiffs who could (and would) tear another animal to *shreds,* many Mastiffs do not possess that driven temperament. But, so, I guess for the sake of the question I'll assume that this is a battle-ready Mastiff, in which case.. well.. I'd still have to go with the hyena. Because I'm a nerd, I know that the typical bite force recorded for the spotted hyena out-measures that of the Mastiff by 400-500 pounds, so I think that the hyena would win (but it'd be close!).
7.) Since the Canadian lynx is known to generally have low stamina, I'd guess the caracal would win.
8.) Probably the snow leopard. I picture the cheetah putting forth great effort at first, but much as they quickly over-exert themselves by running with immense bursts of speed, it would run out of stamina quickly.