No, we don’t use it as an adjective.

From the first sentence of https://en.wikipedia.org/wiki/NP-hardness : "In computational complexity theory, a computational problem H is called NP-hard if...".

Of course, we can’t rule it out as a possibility.

Exactly, we don't know for certain. Of course it's very unlikely, but in math when we say we know something for certain, it means there's a theorem to that effect. There reason to think that factoring is not NP-hard but we don't know for certain. If you still claim otherwise, can you cite a theorem?

Anyway, yes, you're confused, and at this point you're spouting misinformation. You might consider reading a book or taking a class.

If there were polynomial time solutions to that many problems of that degree of importance, surely we would have discovered something by now.

It's still an open problem, there's a $1 million Clay prize waiting for you to claim it if you have a proof either way.