Since gravity on a space station is generated through centrifugal force, what happens if you jump so that you precisely negate your acceleration? Would you hover in place?
“jump so you precisely negate your acceleration” is not a meaningful statement, and there is no way to hover in place. If I’m not ninjaed I’ll come back with a more detailed explanation, but it’s going to take me some time since I have to think about it and draw some diagrams. 
Try imagining it this way: Replace your space station with a nearly frictionless disk on a turntable with a rim, and replace yourself with a ball bearing. As the disk spins up, the ball bearing will appear to stay in place, but will gain speed due to friction and eventually spin with the disk at the same speed.
Now give the ball a short burst of acceleration directly towards the center. A jump. This will give the ball some speed towards the center, but it also has some speed tangential to the disk at the point of and instant of the jump, so it’ll move across the disk creating a chord as seen from above the disk, but hitting the disk close to the point of departure and so seem to be “jumping” as seen from the rim.
If we go back to the space station, we can try to examine the “jump so you precisely negate your acceleration” statement and show how it doesn’t make sense. Centrifugal acceleration is pushing you inwards, so to negate it you must acceleration outwards. That’s what we call falling in regular gravity and would look the same in a space station, and you could do that by pulling your legs up faster than your center of mass falls. You wouldn’t land in the exact place that you were standing, due to the coriolis effect, but otherwise it would be the same.
To hover in a gravity field you direct a force, such as a stream of air, downwards and adjust it till it equals that of gravity. On a spinning space station you could do the same.
Now what if we had an airplane on a treadmill in a revolving space station? 
If you got on a bike and cycled in the opposite direction to the rotation of the station, at such speed as to negate the rotation, you would be weightless (and indeed the bike would start to lose traction as you approached this state).
At the point where you’re essentially staying still whilst the station rotates past you, you could jump off the bike and float up in the air, however, the atmosphere inside the station would also be rotating against you and this would tend to make you start rotating around again, which would make you fall back down to the floor.
Reality check: AFAIK, there has never been a real space station with rotary artificial gravity. Several such modules have been proposed for the International Space Station but it doesn’t look like it will happen any time soon.
Well sure. It wouldn’t be an interesting question if we were already doing it. 
Instead of becoming weightless by bicycle power, I think we should do it the other way around.
As you ride your bike this way around the inside rim, you push the station that way in reaction.
So using human powered bicycles to create the gravity is the way to go. The harder you work, the heavier you get and the harder you have to work.
Maintaining full gee means breaking a sweat now and then.
Although, once you got it spinning it wouldn’t stop.
Maybe switch bicycle directions every few days, spin the station down to zero gee then back up to full weight.
Yes it would. If you set it spinning by reacting against it as you cycle up to speed, you would stop it spinning when you brake.
Action and reaction at both ends - it’s a closed system and you’re inside it, so all of your actions must net out to zero.
I don’t think this is exactly correct.
As you rode the bike you expended internal bio-energy causing the acceleration of the space station, which continually added to the motion of a massive object which then has momentum.
Braking on the bike would drop the rotation speed by some amount, but then you would be stopped relative to the space station which would continue rotating.
It seem like the braking force would only match your pedaling force if your pedaling was for a very short amount of time.
As I was picturing that, it seemed that the braking force was limited to the mass of the rider whereas the acceleration force was only limited by the amount of bio-energy the rider contained, and that he could continue to convert energy to add to the acceleration.
But, I forgot about the speed difference between the rider and the station when considering the braking force.
So ignore my post (I think).
It’s not a matter of conservation of energy, but of conservation of angular momentum. Conservation of energy is always true, of course, but it’s difficult to make practical use of in solving physics problems, because energy is sneaky: It can hide. Momentum can’t hide, though, and angular momentum can only hide with difficulty, so using those to solve problems is much more useful.
So this got me thinking: how to construct a system that allows the person to convert energy into space station rotation while keeping the person’s speed relative to the station at zero.
It seems that to do this, there must be some part of the station rotating in the opposite direction, maybe an inner or outer hub, which means we’re rotating two objects in opposite direction and thus having to expend double the energy (compared to just rotating the station) for whatever ultimate gravity level we are trying to achieve.
Is this correct? Is there any tricky way around this?
Also, it’s only acceleration that really matters, not speed. Once the rider is going as fast as he can, maintaining that speed will not result in any additional reaction against the station - and coasting to a gradual stop gradually undoes all that was done.
This is the concept of a reaction wheel, which is essentially a piece of the spacecraft that is spun to make the rest of the spacecraft spin the other way. Ordinarily they are used more for attitude control than to impart large angular momenta to the rest of the spaceship.
If you want to get around having a reaction wheel, the easiest way is to just use propellant instead. Then the station is not a closed system and its total angular momentum need not be conserved.
You do need to end up with something (a wheel or discarded propellant or whatever) with the opposite angular momentum, but that does not necessarily require spending double the energy. You can do it with an arbitrarily small amount of extra energy, by putting your counter-rotating stuff at very large radius.
I remember an episode of Babylon 5 where they got the physics partially right… they were riding in the monorail down the axis of the station, so they should have been in free fall (although they had gravity in the cars). There was an attack of some sort and the Commander got tossed out of the car, and he drifted slowly along until he got to the station, which was actually moving along at a pretty good clip and was going to squish him like a bug when he ran up against it.
Then an angel saved him.
Some proposed rotating space habitats (e.g. O’Neill cylenders) are composed of two habitable conter-rotating components of similar size and are designed so each component can act as a reaction wheel for the other. If spinning one of these up from zero, it doesn’t absolve you to pump extra energy into your reaction wheel, but it does make it so that energy serves a useful purpose your “reaction wheel” is now living space, too.
Imagine the fun of sports in the space station gym. The results for the high jump and long jump will be quite different depending on the track’s direction. Games involving throwing balls will be something: players will have to learn to compensate for coriolis effects.
It makes me dizzy just thinking about it. I doubt I could watch without getting an upset stomach, let alone play.
There is one other “trick” to spin a spacecraft: magnetic torquers. Which is just a fancy name for an electromagnet used to “push” the spacecraft against the magnetic field of the earth. Or more precisely, a way for the spacecraft to push against the earth through magnetic force. But it’s not very powerful, and it only works in low earth orbit.
Still, it’s a very useful device to supplement a reaction wheel. One problem with a reaction wheel is that if there is even the tiniest bit of external torque on the spacecraft - say, drag from the residual atmosphere, solar wind, etc., the reaction wheel needs to spin faster and faster to maintain the proper orientation. Eventually it will reach a limit. So they need to use a magnetic torquer or thrusters to offload the reaction wheels.