Learn Italian: Nostra nonna è gentile. - Our grandmother is kind.

You are right that Italian usually requires an article before a possessive, like la mia macchina (my car). But there is a major exception: you drop the article for singular family members. Since nonna (grandmother) is a singular family member, it takes no article here.

If a group of brothers is talking, would they say "nostro nonna" instead of "nostra nonna"?

No, they would still say nostra nonna. In Italian, possessive words change to match the gender of the person or thing being owned, not the people who own it. Because nonna is feminine, the possessive must be the feminine nostra.

What happens to this rule if we talk about our grandmothers in the plural?

The rule disappears! The "no article" rule strictly applies to singular family members. If you want to say "our grandmothers", the article comes back and you must say le nostre nonne.

Why is "gentile" used here instead of "gentila" to match the feminine word "nonna"?

Most Italian adjectives end in -o or -a to show gender, but gentile belongs to a group of adjectives that natively end in -e in the dictionary. Adjectives in this group use the exact same -e ending for both masculine and feminine singular nouns.