Hi
The current from your generator is governed by the size of the winding. If the winding will handle 5A then at 5A it will last. Your 5A generator will be capable of delivering more than 5A but not for long, but your 5A generator will put 5A into a 6V battery or a 12 or even 24V battery, it will just need to spin faster. The power of a generator is normally in Watts. so your 6V battery with 1A is 6W, your 12V battery with 1A is 12W. Voltage is measured across the battery, current is measured in series with the load.
As the voltage increases the wire gauge can be reduced for the same amount of work which is why 48V is often the voltage of choice for serious off grid systems.
Where you are where I was about 5 or 6 years ago, I would put a prop on anything that would turn, I have the scars and bruises to prove it. I found out people selling motors on Ebay can be very cavalier with the truth when describing motors to be used as generators. I have a 12V traction motor from a fairground ride under my bench, big, low speed, and powerful, as a generator? Total waste of time. I have motors from electric scooters and go-carts, waste of time. In the end I bought a couple of meters, one reads 15V full scale the other 10A full scale and built myself a test rig, a 12V battery, and several bulbs with lengths of wire on to use as loads. If ultimately you want a 12V system go for 12V straight off.
Brian