There is only one suffix on each of your examples:
(Biz) sakin-iz = we are calm
("Siz" or "Sen" + someone else) sakin-siniz = you are calm (plural or singular formal/respectful)
These are 1st and 2nd Person Plural suffixes for present tense "to be" (a.k.a. Copula): Full chart in Tips and Notes
Hi! Yes, the siz form is used for "formal you" form but, like German (and probably other languages), the formal you also covers the form for "all of you", meaning a group or at least, more than one person. In this sentence, we are talking about "you AND the cat"! So... you and the cat are both calm. You would use the "siz" ending to include the cat. That is my take but i am only a fellow student!