Answer 1

Put the button inside of the background class div

Not to rotate the background, use @property to animate the conic-gradient variable.

https://code.sololearn.com/WCkdCcIAFoo2/?ref=app

Note that currently @property only work on Chrome and Edge browsers.

Answer 2

I'm not sure this is what you're looking for, but try this:
@keyframes spin {
from {
transform: rotate(0deg);
}
to {
transform: rotate(359deg);
}
}
https://techstacker.com/how-to-rotate-images-continuously-with-css-animations/

Answer 3

Your Mom I'm sorry, I don't really get you, how would you use @property to change font-awesome, and what do you mean by "using rotate on the syntax"?