Absolutely not. The starter gear must fully disengage the flywheel gears when you release the starter. Otherwise when the engine starts the starter will continue to be turned by the engine.
It is not supposed to be on when engine is running unless there is a problem.
Because Mitsubishi Eclipse 1996 doesn't have automatic headlights.
Sounds like you might have a bad starter relay.
The starter turns the flywheel at the back of the engine. When the flywheel turns the whole engine turns. The timing chain is on the other end at the front of the engine. It is connected to the crankshaft and camshaft (s). So indirectly yes it does. If the whole engine is turned but the timing chain doesn't, there is a problem. Probably a broken timing chain.
Then why did you replace a perfectly good starter? ALWAYS test old parts and know what you're doing BEFORE you start throwing new parts at a problem. Find out WHY it isn't starting. Does the starter engage and turn the engine? IF it does, then the starter is probably fine. If not, check the starter, battery, electrical connections, ignition switch and neutral safety switch. If the starter has ALWAYS engaged and turned the engine, you should not have changed the starter in the first place; instead, you should be finding out what's wrong.
The actual problem is the starter continues to turn after key is turned off and pulled out of ignition.
Besides the smart alec answer "start an automobile", a starter has a small gear, that will both extend as well as rotate when the key is turned. This gear will engage teeth on the flywheel/flexplate, which is bolted to the rear of the crankshaft. As the starter turns, the engine will be turned.
Is the selenoid getting power when the switch is turned to the start position? If yes, it's not a switch problem. Is power coming THROUGH the switch and to the starter when the key is turned to start? If not, it's a selenoid problem. If yes, it's a starter problem.
When a check engine light comes on, it means that there are issues with the engine. It gets turned off when the person resolves the problem that caused it to turn on.
Replace the starter motor
I had the same problem. I had been using a very poor battery that was almost flat all of the time. All battery connections turned out to be in good order, worried about the engine begining to seize up hbut found it turned freely by hand. the problem turned out to be the starter motor that had broken down due to constant use of a poor battery condition. Please check your battery condition, connections, that engine turns freely and your starter motor. A lot of garages do a start and charge check very cheaply.