Learn Italian: Non vado, che sono terribilmente stanco. - I am not going, as I am terribly tired.

When forming an adverb from an adjective that ends in -le or -re (like terribile or regolare), you drop the final -e before adding the suffix -mente. So, terribile becomes terribil-, resulting in terribilmente.

I usually see perché for "because". Why is che used here?

In spoken and informal Italian, che is frequently used as a causal conjunction. It acts as a quicker, more conversational replacement for perché, siccome, or dato che, and it translates perfectly to "since" or "as".

Can I put the che clause at the beginning of the sentence?

No, when che is used to mean "because", it can only introduce a clause that comes after the main statement. If you want to state your reason first, you must use a different word like siccome or dato che instead.

If a woman were speaking, would terribilmente change its ending along with stanco?

No, only the adjective changes. A woman would say stanca to match her gender, but adverbs like terribilmente are invariable, meaning they never change form based on gender or number.