This post is getting long :/ sorry bout that. I actually started off talking about the q, and then thought about the passive, so to make the order logical I'm restarting here.
First, +1.5% movespeed per stack up to 10 stacks seems high, unless the stacks of Nomad and Predator don't stack with each other (i.e it takes 10 seconds for one to degrade from max before the other starts powering up) in which case it's probably a little underpowered, and I'd suggest capping the stacks lower (~3-5) for balance. Predator also currently gives at lvl 1 a maximum AD boost of 29, 6 more than a wriggles lantern (1200 gold), and with the armpen passive from q is even higher, effectively more like 40 which is about a bf sword (1550 gold). This would be completely OP in laning phase, and even late game when it's giving 80 AD is really overpowered. I'd halve these values, either by reducing max stacks, or by reducing the base damage given to .25*lvl (2.5 extra max dmg lvl 1, 45 extra lvl 18) which is still significant.
The Q ability seems really overpowered, for example Olaf's ult only gives him 20 flat pen at rank 3, gives much better tenacity (but it's an ult, and lasts at most 5 seconds), and maxes out at the same 60 armor/mr pen (at lvl 16 as opposed to lvl 8). Also it doesn't give the rediculous sustain and damage output that feed would give (this is basically warwick's passive on steroids--the passive maxes out around 45 dmg total, out of 15 per auto attack, stacking up to 3 times). Considering that this ability costs no mana/casting resource, and starts off at 60 dmg, it's really broken. Maybe reduce max dmg a lot, OR make it stacking dmg on single target, say 10-15 per auto-attack/ability used max 60. Also, I feel that despite this overpowered q the champion would be under appreciated in high level play because starting in one stance for a gank or team fight means that you can't easily switch to the one that's 2 away from it i.e fight and feed during short engagement fights (ganks/most teamfights pre baron).
My suggestion is to nerf the armpen to 15-20 at max, the max armor/mr (20-30 is probably reasonable), the sustain (make it only occur every 3 hits like pheonix stance, or diana's passive/ahri passive etc), and to make switching between the stances more like TF's pick a card so that higher level play is possible. I would also increase the cooldown at low levels to around 9 seconds, and make it scale down to 4 seconds by rank 5. That way if you lane you can't just trash top lane at lvl 1 and snowball with no items to become GOD by lvl 8. Reduce the tenacity by about 15-20% (it's a lot stronger than Irelia's passive against single targets, or small groups of enemies), and the "upon getting hit movespeed increase" is a little rediculous from flight (since it stacks infinitely with itself). At least give that effect a cooldown (imo 2-3 seconds would be good--you may have other ideas).
Rush is a nice ability, and I think it's pretty well balanced, although I might change the knockback effect to be feed (this is for ganks right? you want that ability on your fight or feed ability), and then do a fear for the flight (or something). If you don't rework the q mechanic to be like pick a card, there's no way to go from knockback in flight to the move speed slow from autoattacks before the enemy champ gets under tower unless they're really over extended, and you have red buff, phage, or exhaust.
Thickening hide seems redundant with the armor bonus from flight. At max stacks you'll currently have over 135 extra mr and Armor just from these two abilities. Also, I thought the idea behind this champ was adaptation? Why does it only have one mechanic? At the least I think eviscerate needs to consume charges on Thickening hide to go with your idea. He should be super tank AND super dps, he should be super tank OR super dps as the player decides to play him.
Claw and Fang is basically a buffed version of Kha'zix ult, except you don't have to expend evolution points to get a second leap (on a different ability I might add), and a second invis (really third leap and invis when you consider Rush). You should break this ult down so that it mirrors your early idea with the champion; adaptation. Extra leaps for Feed, extra invis for Flight, or extra dmg for Fight.
Overall I like the champion idea, but it needs a lot of work.