Question: Is there a reason, that a facility full of monsters would ever, under any circumstances have a System Purge button? Is it just to make satire of the fact that the antagonist is usually stupid?
Answer: As you said, it is a satire of the common Deus Ex Machina trope you find in such films.
Question: How did the demolition woman who is shown on a monitor shooting herself get the gun in the first place? After all, she isn't a security guard.
Answer: With all the monsters being kept in captivity, it would make sense for the facility to have an armory as well. Guns may have been handed out once the cages were opened.
Answer: It's stated that the tunnel should have blown hours ago, so the initial failure is most likely human error, that the order to blow the tunnel simply didn't get through to the demolition team on schedule. The subsequent problem blowing it up is referred to as being related to "a glitch up top", later stated as a "power re-route upstairs". While not stated outright, the only plausible explanation is that this was, as you rightly stated, due to Marty messing around with the equipment that he found in the hatch.
Tailkinker ★