By the time Mirror Realm enters the realm, it's already a copy. Thus, it can copy a nearby site, and then promptly loses all abilities of that copied site (but still is the copy, without abilities). See 'Apply Ongoing Effects In Order' in the codex for nitty-gritty details.

What if Mirror Realm loses its abilities after it enters play (e.g. Smokestacks of Gnaak is played later)?

Mirror Realm remains a copy, but any abilities that were printed on the original are now ignored.