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"- Periodically (every 10 minutes or so), the game will minimize itself and dump me out to the desktop."
Are you hitting any escape key combo's? (ALT+Esc, Windows Key, etc). You sure you have not picked up any spyware / adware in the last few days?
"- When I get the Aethric Mirror (or whatever that device is called), and use it to jump to inhabited worlds via the command "mirror", I often get the message "you have won a decisive victory""
The enemy fleets were guarding the planet, you got to the planets surface because after using the mirror there were no enemies there (because you switched places with the enemy force). Technically its a victory. You should only see hostile creatures at the planet again if you use the mirror from that planet to switch places with another fleet, and then go back to the planet with standard engines.
"- This is just a small thing, really, but I notice that while the mission types all seem to be very similar, the military one seems to be easiest simply because the ship has more slots."
On the military missions, you get points by destroying enemies and getting equipment, its far more dangerous if you dont know when to engage and when to leg it. The science missions you can get a good score even if you never engage enemies. I dont know that one is meant to be harder than the other, they are different approaches to the game, and you happen to be inclined to the military method.
Hope that helps