Learn Portuguese (Portugal): A nossa família mora em Lisboa.

In European Portuguese, it is almost always mandatory to put a definite article (words like o or a, meaning 'the') before a possessive word (like nossa). Since família is a feminine noun, we use a. You are literally saying 'the our family' (a nossa família).

Does nossa change depending on the gender of the people speaking?

No, it only changes based on the word it describes. Possessives must match the gender (masculine or feminine) and number (singular or plural) of the thing being owned. Since família is a feminine, singular word, we use nossa, even if the people saying 'our' are all men.

Since a family is made up of multiple people, why is the verb mora in the singular form?

Even though a family has many members, the word família is a singular noun (one family). Just like in English, you treat it as a single unit, which is why we use the singular 'he/she/it' form of the verb morar (to live), which is mora.

Why is it just em Lisboa and not na Lisboa?