1, Yes, their are various ways to split the Oxygen atom from Hydrogen Atoms that make up water, but all are high in energy costs. Furthermore remember it takes as much energy to spit the atoms apart (by this I mean splitting the Oxygen atom from the Hydrogen atoms) as you get when you "burn" the Hydrogen in your car (And re-unite the Hydrogen atoms to the Oxygen atom). There is no free lunch. Please note the above assume no "Loss" and you will have some sort of "loss" in both re-actions.
2. Hydrogen is the smallest atom know to man. In the gaseous state can be contained for the Gaseous state of any atom is larger then when it is in the Liquid or solid form, thus we have material that can contain Hydrogen gas with a good bit of efficiency. On the other hand, when you compress Hydrogen (Which requires additional energy) to a liquid form (So you can have Hydrogen in your "Gas Tank") the resulting container will leak Hydrogen at about 1% per day. This leak is right through the container wall itself, not the seals as with gasoline but right through the container walls. It is for this reason Hydrogen is produced and sold locally, after about three months is is almost all gone (Yes I know as Hydrogen leaks you have less to leak out so the am out of Hydrogen lost drops over time, but the rate of drop is constant thus I am using 90-100 days to say all the Hydrogen will be gone. Much of the hydrogen will still be in the Container but no where near what it was 100 days before).
While most people do NOT keep fuel in their car for more then two weeks at a time, your local gasoline station does. Expect a huge loss of Hydrogen after it is produced. This loss, like losses as part of transferring from the storage tank to your car, will have to be added to the cost of the Hydrogen you end up paying for.
3. Do to the fact Hydrogen is a Gas at normal Temperatures and pressure, (Hydrogen only becomes liquid at 20.28 K, -252.87 °C, -423.17 °F at normal pressure), to be store-able Hydrogen must be pressurized till its Temperature and pressure is below its "boiling point". This requires a very strong and heavy container (and its weight has only a minimal affect on Hydrogen leaking through the container walls) which requires careful handling AND additional energy to be moved around as part of your car.
More on Hydrogen itself:
http://en.wikipedia.org/wiki/HydrogenRepeating an old Joke (Modified to apply to Hydrogen) "Hydrogen is the energy of tomorrow and always will be". Hydrogen has a lot of things going for it. Can be produced out of water, can be produced almost anywhere. Has the most energy for any chemical source of energy (Nuclear provides more power per pound but that is do to Nuclear frisson or fusion of atoms themselves NOT the energy connecting atoms to other atoms). The problems are first there are NO nature source of Hydrogen, it has to be made (Can be made anywhere BUT takes energy to be made), Second the high loss rare do to the fact it is impossible to store quantities of Hydrogen in any form other then the Liquid state AND then even in the liquid state almost impossible to keep in storage for any length of time. These are problems inherent with the use of Hydrogen and no one has addressed them for under present scientific theory there is no solution. Now if a cheap way is found to convert water to Hydrogen and so so when the Hydrogen is wanted then most of these problem can be avoided but that means a access to some sort of power at every filling station that would equal the power from you local power station (i.e. your local power station where your gasoline station is today). Possible? Yes, probable NO. That is why people like me foresee a fundamental change in out social structure, a move back to the inner cities do to abandonment of the Automobile as opposed to the various attempts to preserve the automotive culture we Americans have adopted since the 1920s.