As Dan said, precession is the slow rotation of a spinning object and is a part of gyroscopic motion. Precessision is caused by an external moment being applied to the spinning object (out of axis with the spin). There are no axial forces involved.
There are actually no forces acting axially on the clutch (along the axis of spin), other than caused by snowmobile cornering forces. It doesn't matter whether the clutch is on the right or left side of the sled, the forces are effectively equal - depending on which direction you're turning. As you are turning there are gyroscopic forces on the engine bearings, but they are radially applied to the bearings.
The right hand rule is simply a method of defining a vector that relates to rotation, an applied torque (or moment), or anything else related to rotation (rotational velocity, acceleration, etc.). The rule is that your four fingers of the right hand are wrapped in the direction of the rotation when your thumb points in the direction of rotation. There are no axial forces along this vector, just rotation, etc. in the finger direction.
As far as efficiencies. A typical snowmobile transmission is between 75% and 90% efficient, depending on the gear ratio and clamping forces on the belt. What robs hp is bending the belt around a small radius (converts hp to heat - more at low and high gear ratios), belt slippage and creep, clamping forces create friction (with current designs the clamping loads are higher than required for the engine torque at high snowmobile speeds), and air drag on the clutches and belts. I can't remember the numbers for a chaincase off hand, but they are considerably more efficient (I believe 95-99% efficient). A constant ratio belt drive is somewhere between a variable belt clutch and a chain (depending on how well it's designed). One big advantage to running a belt instead of a chain is a belt removes much of the harshness and harmonics from the system and may be able to allow lighter shafts, etc. to be used (or less vibration for the driver, longer lasting parts, etc.).
A lighter clutch would reduce rotational inertia allowing faster accelerations, but won't effect (significantly) constant speed efficiency.
A snowmobile is quite an ineficient peice of equipment really. By the time the engine hp reaches the ground it is cut approximately in half. I'm sure if you hunt around a little you could find real numbers, but if you have a 100hp sled, you loose on the order of 10-25 hp in the transmission, 2-4 hp in the chaincase, and another 10-30 hp in the track drive, slide friction, bearings, track flex, etc. After that it takes a fair bit of hp to pack the snow down under the track and push the track lugs into the snow, so what you're left with moving the machine forwards is probably around 30% of the total engine hp for good snow conditions.
Keep in mind these numbers above are educated guesses. I haven't worked out the actual numbers. The transmission efficiency numbers are from some testing that was performed by AAEN Performance during the developement of some of their racing transmissions (they actually quoted 80% to 92%, but I believe a typical snowmobile is slightly worse).