It should be kept strictly professional.
Interaction between equals or co-workers is something different. But between teachers and students the balance of power is so very unequal that keeping things professional is the best way to go about it.
PS the answers are yes, yes, it depends and it depends.
Thanks for the cordial reply. At nearly seven decades old my thoughts and words toward the opposite sex place me at a much younger age. It’s not easy being human.