Well, if you want to be a cripple...
seriously?
1) humans are neutrally buoyant in water - meaning gravity doesn't push us down enough for us to get decent traction in it. in fact most people naturally float in water and it actively pushes us up off the ground like a helium-filled-balloon. (meaning moving through water example is invalid unless we're talking about super-thick atmospheres)
2) Lyra moves on all fours when she wants to move fast. (has paws on ground higher % of time during motion than bipeds)
3) Legs apply horizontal force with each step they take, for the duration of that step.
4) air resistance applies the same amount of force for a given speed no matter my weight
5) I apply the same amount of force against that air resistance no matter my weight, an increase my total momentum by the same amount (unless I become literally as light as or lighter than air, or close enough to it that my terminal velocity would be measured in inches per second - again, buoyancy matters)
6) I am not a projectile - my feet and claws remain in contact with the ground and continue to apply force as I move.
7) at the kind of speeds air resistance and my reduced momentum for a given speed - not a given amount of force applied to my body, for which the momentum would be the exact same no matter my weight would actually matter, I'm already going much faster than normally possible.
you're right that for a given speed, air resistance will decelerate me faster, but you are wrong because for a given speed, air resistance still applies the same amount of force, and will only reduce total momentum by the same amount in either case.
You will only become correct if I find myself in a thick enough atmosphere that I literally have trouble staying on the ground, or if I am in free-fall.
This is the same point I said before, just a little more broken down for you since you haven't actually addressed it.
1) This is correct, but it doesn't mean you'll run faster, it means you will have trouble going as fast, because each step will send you up unusually high, and you can't push off the ground if you're airborn.
2) I'm no cat expert, but based on your speed, I'm guessing Lyra moves like a cheetah would when she's running. Watch a video of a cheetah running full-speed: it usually is airborne, and rarely touches more than one foot to the ground at once. If you're less dense you'll just go flying higher into the air with every step, so you'll be subject to more air resistance, and be less able to resist it due to your lack of weight.
3) No, they apply a force identical to and opposite to the force they apply on the ground, which is primarily upwards, and somewhat forwards. You can't get a completely horizontal force if you aren't pushing of a vertical wall.
4) This is true, but remember that any force will push you around more, because you have less inertia due to your decreased mass.
5) No, you don't. Force equals mass times acceleration. You have less mass, and I don't think you go fast enough to counter this. I doubt you go twice as fast as normal, or any faster at all for that matter.
6) If you run like a big cat you'll spend less time in contact with the ground.
Watch this.7) Air resistance matters at any speed, if she's running. Because of the way cats tend to run, Lyra is going to move slower than she normally would if she tries to sprint, not faster. She's not going to look like a sleek and deadly tigress, she's going to look like a furry version of Neil Armstrong, bounding up and down slowly, easily blown off course by a stiff breeze.