I don't think that's a satisfactory answer for a layperson. Infinity is a size/quantity, for most people that usually means "number". The reason it's not a number is because it doesn't exist on the real number line, but again I don't think the helps people intuitively understand what's happening.
I think it does. A layperson scratches their head when trying to think about how to circle the square of "infinity + 1" or "infinity / infinity". Framing infinity as something outside of arithmetic is a starting point for a more interesting discussion of countability and types of infinities. Infinity is bigger than any number, so then how could it be a number?
Um, duh, you tip an 8 over. If it's not a number, why is it made out of a number? /s
There are numbers that aren't on the real number line. i, for one. Omega, for another.
It’s fairly easy to extend R with plus and minus infinity. It’s called the affinely extended real number system and you can extend the usual operations to somewhat work in it. It loses most of the interesting property of (R,+,x) but keeps enough for arithmetic to work mostly as expected.
Numeric operators only work on numbers. Operators can work on all sorts of mathematical objects.