Is it true that, according to the criminal law in the USA, it's illegal, in a trial, to persuade a jury that something happened (or not) by using exclusively facts not in evidence?

You have to have proof. It’s the basis of the judicial system.

You have to have evidence, and prove to the jury that what you are alleging is true ‘beyond a reasonable doubt.”

Not exactly.

Court rules don’t characterize pleadings as “illegal” or “legal”, they are either allowed or not permitted. Lawyers rarely go to jail for how they argue cases. Their statements are stricken and juries are ordered to disregard whatever the judge determines is improper.

But defense attorneys in particular are allowed fairly wide latitude to present an alternate theory of the case which may suggest that someone other than their client may be guilty. This especially true during opening statements and cross examination of witnesses.

Not exactly.
“circumstantial evidence… (1736) : evidence that tends to prove a fact by proving other events or circumstances which afford a basis for a reasonable inference of the occurrence of the fact at issue” -from Merriam-Webster’s Collegiate Dictionary, 11th. ed.