I suggest taking your dog to your veterinarian immediately. Dogs have bacteria in their mouths (just like humans), so your dog probably has a bacterial infection that will require medication to get better.
In my dogs it is an invitation to play.
well some dogs can do paw stands if there owner holds there feet
If it licks its paws more than usually, it might have a sore paw, an injury, an insect bite, a splinter (sometimes it's glass and it can be difficult to spot it).
the cats paw lets it balance and the dogs paw doesnt
They rarely get along. It depends on your dogs' personalities, if it's good, sweet, calm, patient, doesn't paw, doesn't growl, doesn't bite, doesn't chase cats, and the cats' personalities, if it's good, sweet, calm, doesn't claw, doesn't hiss, doesn't bite
no
clean it
rub peanut butter on the dogs paw dah
Yes dogs have paw preferences most are left pawed because most humans are right pawed and when you train them they do the opposite
Yes, they expel heat through their paw pads.
maybe you can paint it if the dog dosnt move their paw
There are too many variables to answer this question. Some dogs might be hiding something, another dog might have a hurt paw, and another dog may simply like to sit this way. It would depend on many factors.