If you have a new battery, but it has discharged after a ride, then 99% sure it's a charging problem, not a battery problem.
It could be a short as swest suggested, but in my experience, 90% of the time it's the regulator/rectifier as just blown. I think you would have a generator, so no rectifier for you - my bet is the regulator. They dissipate a lot of heat, and do wear out.
Get a multi-meter - they're cheap and you will get your money's worth the first time you use it.
Back to the jumping-bike-with-car debate, I see no reason why you couldn't. The way they work is pretty much the same, as is the voltage. In fact, a new car generally has a lower charging voltage than an old one - to preserve the battery - and probably lower than your bike. However, I don't think it's worth the risk - but also, there's no need to start the car first. A car battery by itself will start a motorbike over and over again, and still start the car no problem. Just don't start the car, and jump from the battery.