It may have done some damage to anything that runs at all times.. Wireless locks, radio, computer systems and the Alternator could all have been hurt. The length of time the battery was on wrong makes a big difference as there is some protection on all this equipment.
I would check the weiring harness from the car to the starter
You may have installed the thermostat backwards.
it will arch out and melt the knobs on your battery and your cables
Car battery charges can check the damage done to the actual battery but to the entire car is another story
once a battery is connected even if the key is not in the ignition, there is always power at certain points in your car. Connecting a battery backwards will not start your car, but it can and will fry the wires to your starter.
Turn it the other way.
As long as the cables are connected properly.
Yes they can be installed in any car or truck that has a good battery.
When you connect the battery backwards you can cause major damage. You can blow the alternator or its fusible link. You can destroy electronics, particularly the various computers - and these are not cheap.
If the alternator is installed incorrectly, yes, it can drain the battery.
Replace the wiring that was burnt.
If the remote car starter stopped working after a new battery was installed in the remote, the remote needs to be reprogrammed. If the remote starter stopped working after a new battery was installed in the vehicle, the computer might need to be reset by disconnecting the battery for a few minutes.