Would an arduino and relay board kill a car battery?

A schematic drawing would be a big help. As far as fuse, you will have to determine that by the sum of all the 12 amps being used by the equipment. If ANY of your equipment is connected to the automobile chassis, and you have a wire from the battery - (negative) terminal, you need to also fuse that negative wire with the same size fuse as the positive.

Are you going to start the engine with this equipment connected? You may get transient voltage pulses from the starter motor, or you may get voltage drops to 8 volts or so, or both. Best to have the Arduino and other boards powered off while starting the engine. On many cars, the electronic equipment is turned off at the key switch when the engine is started, then back on when the key start is released.

Unless you have a good car battery, the charging voltage may exceed 14 volts. My Diesel truck has exceeded 15 volts, (160 amp alternator) when the batteries were about about shot.

A schematic of the connections, please.

Paul