Yes, exactly - same Germanic root. "Fast" in Scandinavian languages means firm/fixed/stuck, which is also the original meaning in English (as in "hold fast", "steadfast", "fasten"). The "quick" meaning in English is actually the newer one, derived from the idea of being "stuck" on a course.

Yes. And as in fastness (which would be «fastheit» in Nynorsk): https://www.etymonline.com/word/fastness