Try to provide more info. Like does it whine while turning, only going straight. Only in gear, only at some given RPM. Etc.
Remember, all that stuff is in your sub-conscience but we dont have a clue from our little keyboards.
For Keyless, most installers seem to tap into the lock wiring in the driver side rocker panel. If you get down there and see the lock wires tapped into (with a fuse tap or a soldered Tee), that suggests factory keyless. If you see that two wires were cut and legs of wiring run from each cut wire, that suggests no factory keyless or a retard installer. Your dome light thing suggests keyless to me.
Good luck.