Yes, these are possessive pronouns. And yes, they agree with the noun, just like adjectives.

examples:

моя книжка (my book, книжка is feminine);

мій ніж (my knife, ніж is masculine)

моє дитя (my kid, дитя is neuter)

So:

my-your(singular)-his-her-our-your(pl)-their NOUN:

моя-твоя-його-її-наша-ваша-їхня книжка

мій-твій-його-її-наш-ваш-їхній ніж

моє-твоє-його-її-наше-ваше-їхнє дитя

The third person (singular) is the same for all genders.

Also, do non-possessive pronouns change?

They do change according to grammar cases, gender and number, but I can't think of other changes.