Aside from taking it, out there is not much you can do other then give it a spin to see if it is moving smoothly.
You should check and adjust the heating temperature, then turn on the cold water to fill the tank.
Replacing the water pump will not cause the check engine light to come on. Either it is a coincidence or you have accidentally disconnected a wire or damaged a sensor somehow.
pull a plug check for spark if spark check for fuel dont think merc jet should be run out of water
Check the water pump drive shaft. If water is leaking from the hole beneath the drive, it's definitely a bad water pump seal and you can't repair them.
Remove all coolant from engine and do not drive car anymore!
Replacing the water pump and belt should not affect turn signal operation.
A water pump should have nothing to do with the bearing in your engine. If you are hearing a knocking from the engine then you are going to need to replace/rebuild. I have seen bad waterpumps make knocking nosies but you said it was new. Id check your work!
Check the radiator fan operation.
It's recommended to do when you are replacing the timing belt.
It is not recommended, overheating can cause severe damage that cost a lot more than just replacing the pump.
Replacing a bad water pump can help to stop a motor from overheating. Hopefully that was the cause of the overheating. If it does not stop the overheating, try replacing the thermostat. Make sure you have the proper mix of antifreeze and water, the radiator filled with it. Check the level of coolant FIRST. Next, the hoses, and third the thermostat. Then, disconnect a hose and see if, by starting the vehicle, if there's water pressure. If not, its water pump time!
if this is front wheel drive it will be on the passanger side bottom of engine good luck replacing that it it is rear wheel drive its in the front bottom a little eaiser