Yes, you can connect it directly to the battery. You should connect it to a fused battery terminal to be safe. If you slip and short something, or if the motor is already shorted you could get some sparks and if you're lucky some welding. LOL
Test the relay by connecting the coil to the battery (again a fused line). You should hear the contact click firmly.
ok I connected the pump to the battery while the back of the pump was open. The pump is functional but I can see sparks flying in the back of the pump. Is that a sign of a bad pump ?