What are assistant professors called in North America?
In North America, one would only mention an academic's rank (assistant professor, associate professor, full professor, etc) in formal contexts where it is important to describe their exact job title (in a CV, business card, news article, etc).

Otherwise, the word "professor" is used generically to refer to any tenure-stream university faculty members, and in some cases also non-tenure-stream.  For example, "Susan Jones is a biology professor at Harvard."  That would still be correct if her rank is assistant professor.  Or: "The conference was attended by 300 professors from across the country."  They need not all have been full professors.

It is also the word used to address any of these people.  ("Professor Jones, I thought your paper was very interesting.")

"Doctor" is also used as a term of address, assuming the faculty member in question holds a doctoral degree.  ("Doctor Jones, please tell me more about your experiment.")  Whether "Professor" or "Doctor" is to be preferred is a matter of local custom which varies from one institution to the next.  But "Doctor" is not used to refer to the job itself.  You would not say "Susan Jones is a biology doctor at Harvard."

In North America, titles are never stacked.  One does not say "Professor Doctor Jones".