Does "Mnoho" require the genitive? Ergo, the difference between Mám mhono sestry and Mám mnoho sester?
Yes, it requires genitive. The same with málo, hodně etc. Mám mnoho sestry is incorrect.
So "sester" is genitive plural of "sestra"? And why is it like that? Is it an irregular? I haven't found any rule regarding such case
It is. See https://cs.wiktionary.org/wiki/sestra for the full declension of sestra. You might want to add this site to your learning resources.
Thank you, BhBass, that site helps me so much!