The project you suggest is definitely possible. In fact, a few companies have produced prototype hydrogen internal combustion vehicles (see
http://www.hydrogenhighway.ca.gov/vehicles/ice.htm).
Now for your questions:
1. What would be the best cathode and anode?
I don't know enough about the subject to suggest the best materials, but I've heard graphite is good (and easy to obtain; just shave the wood off of the outside of a pencil). Unlike using metals for electrode, a graphite electrode will not degrade as much over time.
2. How much oxygen and hydrogen gas will result from the water, say one gram? and how much voltage is required for one gram?
Well, since you are taking a chemistry course, you should be able to write a balanced chemical reaction and calculate the mass of oxygen and hydrogen produced (if you want to calculate the volume, just use the ideal gas law).
A note about electricity. There are a few terms that get easily confused. Voltage refers to the difference in potential energy between the cathode and anode. Current refers to the amount of electricity flowing through a device. Charge refers to the total amount of electrical energy held by a battery. As an example, you can consider, the common battery types AA and AAA. Both put out 1.5 volts and the only difference is the amount of charge they carry. So, AA batteries would power a device longer than the same amount of AAA batteries. For devices that require a bit more kick (i.e. more voltage), there are 9 volt batteries or the wall socket (120 V in the US).
Anyway, the rate of hydrogen production from electrolysis is depends on the amount of power (current multiplied by voltage). The exact amount of electrical energy to produce a certain amount of hydrogen depends on how efficient your electrolysis setup is.
3. Do solar panels keep a constant stream of voltage so long as the sun is shining?
Here I think you mean current, but I don't know enough about solar cells to give you an accurate answer. I would guess that the voltage and current stay fairly constant given a fairly constant stream of light hitting the panel, but I'm not sure.
Finally something to consider. One of the big advantages of hydrogen fuel is that hydrogen can be used in fuel cells which convert fuel into energy much more efficiently than internal combustion engines. In internal combustion engines, most of the energy contained in the fuel is lost as waste heat. Fuel cells lose much less of the energy from the fuel as heat. Of course, fuel cells are much more expensive than internal combustion engines, so that has been one factor (of many) holding back fuel cell technology.
But, good luck with your studies. Changing the way the world uses energy is one of the most important problems facing humanity this century, and chemists will definitely play a big part in solving this problem.