Both words, “extremist” and “right-wing”, have no real hard definitions.

No, right-wing ideology has a very hard and clear-cut definition - all politics that protects power and privilege. It really doesn’t get any simpler than that.

Is being socially conservative right wing?

It’s not supposed to be… but the only people self-applying the term in the US are fascists.

Is supporting capitalism right wing?

Yes. Period.

anti-racist and pro-trans left wing?

That’s not progressive - that’s radicalism.

Is socialism left wing? Is only communism left wing?

Yes.

Is someone an extremist if they condone violence?

No.

Is someone an extremist if they seek to change the system fundamentally?

No, that’s radicalism.

Is someone an extremist if their political beliefs are very strongly held, no matter what they might be?

No.