Yes, it most certainly can.
It is one of hundreds of things that could turn the check engine light on.
Check your battery for loose connections and check your alternator.
You could have the start of a head gasket that is leaking. I suggest that you have a shop to add leak detector to the cooling system and check it with a black light.
Nope
No, but it could cause the check engine light to illuminate.
no but in most late models your check engine light come on
No, a loose gas cap can cause the check engine light to come on but not the low fuel light.
It is possible. A more likely cause is the gas cap was replaced incorrectly or left off after getting gas
Not unless you loose enough oil to cause a lack of lubrication.
You can't exactly pinpoint what caused the check engine light to come on without retrieving the trouble codes. Take your vehicle to AutoZone or similar auto parts store that offers free diagnostic scans and have the trouble codes retrieved and explained. But first... check your gas cap. A loose gas cap can cause the Check Engine light to turn on. Really.
The most common cause of a dome light staying on in a Mercury Sable is electrical issues. If a fuse becomes disconnected or is loose this can cause the light to remain on. Check the dimmer switch and the plunger switches in the door jambs.
You have a blown head gasket