Can you be a dancer without being able to remember any choreography?

I'd say yes. Plenty of folks like to dance their own freestyle.

But 'good dancer' as in professional? Not likely. But I bet there are pro dancers who have to work harder at remembering routines than others.

Sure, dancing doesn't have to be choreographed.

When you say [b][i]"be a dancer"[/i][/b] do you mean professionally or for fun?
If it's the former, highly unlikely, especially if they're part of a troupe or ensemble.