With DC it will turn part of a turn till the fields align and lock in place. It cannot complete a full revolution. With AC it needs to be spun up to synchronous speed, than apply the AC and it will keep running, locked to the supply. This is a synchronous motor. The alternator in a car is different, as it has rectifiers to convert the alternating current produced into DC for the battery etc. If an AC supply is fed to it these rectifier diodes prevent it from working as a synchronous motor. A car alternator will not work for either, without modification.
No, not easily, at least not for single phase AC. Your alternator is a DC powered rotor coil in a three phase AC coil stator. The rotor needs DC current to produce the mag field. The brushes that power the rotor are fixed rings, they don't reverse polarity on rotation, so no DC motor action. Your rotor is not wired for induction nor are your stator coils, so no single phase AC motor action. You need a rotating magnetic field to make it work, you need three-phase AC supply and power the stator coils separately and a DC supply for the rotor. Then you will have a 3 phase AC motor. Easier to just buy a motor, though there are instructions on the internet for doing it. Google alternator as brushless motor
In theory it would work. Rotational energy in an alternator turns into DC current. So theoretically, if you put DC current in, rotational energy should come out. Just like spinning an electromagnet inside of coils of wire makes electricity, putting current through the wires rotates the electromagnet. Good luck!
