When I ride my bike, does half of the energy go into the earth because of Newton's third law? Does the energy in the earth transfer into heat upon my brakes when I utilize them?
If the earth was 2000 times less massive than it is, and I applied the same energy to my pedal, would my bike move (relative to an observer not connected to earth) at the same speed as it would if the earth was normal?
I think if you think about it these two questions are the same.
No, because even though the force that you exert on the earth is equal and opposite to the force it exerts back on you, you're not doing the same amount of work on the earth as the earth on you. Your kinetic energy increases due to the work done by the earth on you. Remember that $W = F \cdot d$; your bicycle moves a lot due to this force, but the earth doesn't really move much at all.
Another way to think about this is in terms of kinetic energy. $\mathrm{KE} = \frac{1}{2} mv^2$, so if your velocity is high, so is your kinetic energy. The earth's velocity is low, and so is its kinetic energy. So the forces are equal and opposite, and the impulse, or change in momentum, is too, but the kinetic energy stays mostly with you.
I think you are confusing momentum and energy.
If you ignore air friction then it would take almost no energy to cycle in a vacuum, just the energy to overcome friction in the wheel bearings (well that and the inability to breath)
Cycling at reasonably quick speeds most of the energy goes into aerodynamic drag, that is moving the air out of the way, and so ultimately to heat in the air.
Yes - when you brake all the kinetic energy goes into the brake pads which is then dissipated into the air as the pads heat up
