Learn Italian: Sono a mio agio in questa casa.

The phrase essere a proprio agio is a common Italian idiom meaning to be comfortable or at ease. The word proprio (one's own) changes to match the subject. Since the subject is io (I), it becomes a mio agio.

Could I say Sono a proprio agio since this lesson is about the word proprio?

No, you must use mio here. The word proprio is only used to mean his own, her own, or their own when the subject is in the third person, or in impersonal sentences. For the first person (I), you always use mio.

Why isn't there a definite article before mio? Shouldn't it be al mio agio?

While Italian usually requires an article before possessives (like la mia casa), it is dropped in certain fixed prepositional phrases. A mio agio is one of these set phrases, similar to a mio parere (in my opinion).

Just to review, why do we use questa here?

The demonstrative adjective for this must agree with the noun it modifies. Since casa is a feminine singular noun, you use the feminine singular form questa.