division with 0 is completely different than multiplication with it because it has no multiplicative inverse. your use of equal signs is flawed, especially in your statement x=0/0 where x is not a set, but an element of it, implying that there is some set such that all x in R is a solution to 0/0 implies that 0/0 = x1 = x2. x/0 is not well defined for any x.
also, there are also equations with other finite amounts of solutions. as such, your statement that there are only three possibilities for the size of the set of solutions leads me to believe that you may have skipped over algebra 1 in your math studies. e.g.: quadratic, cube roots.
theoretically, there could exist a polynomial equation of infinite degree with an infinite number of solutions, such that not all x in R solves it.