Learn Dutch: Ik ben ziek, dus ik blijf thuis. - I am sick, so I am staying home.

Huis just refers to the physical building ("a house"). Thuis is an adverb that means "at home". Whenever you want to talk about being at your own home, going home, or staying home, you must use thuis.

The translation says "am staying". Why isn't there a word for "am" in ik blijf thuis?

Dutch rarely uses a direct translation for the English "to be + -ing" format. Instead of saying "I am staying", Dutch simply uses the basic present tense. Therefore, ik blijf translates to both "I stay" and "I am staying".

Does the word dus (so) change the word order of the sentence?

In this sentence, it does not. Dus acts as a linking word (a coordinating conjunction, just like en or maar), which means it simply glues two normal sentences together. The second part keeps the standard Subject + Verb order: ik + blijf.