Learn Portuguese (Portugal): Tenho frio em casa.

In Portuguese, physical feelings like being cold, hot, hungry, or thirsty are expressed using the verb ter (to have). You are literally saying 'I have cold'.

If a woman is speaking, does frio change to fria?

No, it stays as frio. Because you are using the verb ter, the word frio acts as a noun ('the cold') rather than an adjective describing the speaker. Nouns keep their own fixed gender regardless of who is speaking.

Why does it say em casa instead of na casa?

The phrase em casa is used specifically to mean 'at home'. If you used na casa (in the house), it would sound like you are pointing out a specific physical building, often belonging to someone else.

Is it wrong to add eu (I) at the beginning of the sentence?

It is not wrong, but eu is usually left out. The verb form tenho already tells the listener exactly who is doing the action. You only need to add eu if you want to emphasize that you are cold, in contrast to someone else.