Learn Portuguese (Portugal): A professora tem muita paciência.

Because paciência (patience) is a feminine singular noun. When muito is used to mean 'much' or 'a lot of' a noun, it changes its ending to match the gender and number of that noun. Since paciência is feminine, muito becomes muita.

Is there a trick to knowing that paciência is feminine?

Yes! Portuguese words ending in -ência or -ância are almost always feminine.

Which verb is tem from?

It comes from the irregular verb ter (to have). Tem is the form used for ele, ela, and você (he, she, and you). Since a professora is a 'she', we use tem.

Does professora specifically mean a female teacher?

Exactly. A professora is a female teacher. If you were talking about a male teacher, you would drop the 'a' at the end and say o professor.