Does toothpaste with cool mint cause any allergies on the chin and lower part of face?

Not usually. Mint can be irritating for some people. Generally toothpaste preparations aren't a problem for the skin.
Possibly. If the toothpaste came in contact with those areas, then it is contact allergic reaction.
Yes, it can. . Allergies vary from person to person. If you suspect it is your toothpaste, try something without mint and see if it clears up or comes back.