No the current coming form the battery is still 12V/60A but if the appliance on the end that is drawing current, for example a car radio, exceeds the 10A the fuse will blow. This is usually done because the wire is not large enough to carry 60A and would start to burn or melt if too much current flowed through it. You could hook the radio directly to the battery with no fuse in the circuit and it wold work fine, the fuse is just there to protect the wiring for the most part.

The fuse will only disconnect the circuit from the load if the amp draw is more then it's rating. It does not limit the battery or current in any way. It will in fact allow a 60 amp load through but it will blow out almost immediately.

The fuse only limits the amount of power that is allowed to pass over the set of wires from the battery to the device. It doesn't affect the battery output.

No the fuse is a safety device that limits the current. If the load exceeds 10 amps the fuse will blow. Fuses are there to protect wiring. They blow before the wiring catches fire.

For instance, if I have a car battery with 12V/60A and I put a 10A fuse on it, does it mean the electricity coming out of that battery isn't 12V/60A but 12V/10A ? I'm sorry for the dumb question, I don't know much about this kind of thing.