Learn Spanish (Latin America): Nosotros tenemos dos perros, siete gatos y ocho pájaros. - We have two dogs, seven cats, and eight birds.

No, they stay exactly the same! Numbers from 0 to 15 don't change to match masculine or feminine words, with the single exception of the number one (uno, which changes to un or una). Whether you have male cats (gatos) or female cats (gatas), it is always siete.

Do numbers like siete and ocho need an -s at the end because they refer to plural animals?

No, counting numbers don't take a plural -s. The number itself already tells you there are multiple things! You only make the noun plural, which is why we see perros, gatos, and pájaros with an -s at the end, but the numbers stay as dos, siete, and ocho.

Is it okay to leave out Nosotros and just start the sentence with Tenemos?

Yes, that is completely fine and actually more common! The verb tenemos already means "we have," so the "we" is built right into the word. Using Nosotros here just adds a bit of emphasis, like saying "We have two dogs (even if someone else doesn't)."

Why does the word pájaros have an accent mark on the first "a"?