Welcome to the game and to the forums!

Glad to hear you're enjoying it. Particularly enjoying being beaten
The idea is for it to provide a stiff challenge at whatever level of play someone's at. Some people prefer a challenge such that they just barely win, others prefer a challenge such that they just barely lose; either way you should be able to find it here. Others just like to romp over it without much chance of loss but that doesn't require much careful balancing on our part.
golems, dyson sphere, hero unit and spirecraft turned on. I tried to select all those options that seemed likely to add interest, while leaving me fighting the AI rather than dealing with swarms of marauders or other randomised factors.
Yea, that will still have the AI as the opponent. Is that golems-hard and spirecraft-hard, or what? Both on hard would be quite painful if you're not very used to the game.
After game two I read quite a bit on the forums about keeping AIP low and opted to leave as many enemy planets as possible, having killed guard posts and warp gate but not command station to neuter them. I figured that this would avoid the 15 AIP for each CMD, spread out enemy reinforcements and finally leave me opportunities to leech/zombie farm on a regular basis.
Yes, that's a smart play as long as you don't wind up spending too much time on the neutering (either running up the clock too high, or just getting bored).
I am now on 696 AIP
Whooooa, that's not "keeping AIP low"

300 to 400 is fine on 7/7 (in my personal recent experience, at least) but pushing 700 is.. well, pushing it.
taken 50 of 80 planets
Good grief. It is possible to win 7/7 by taking all/most of the planets, but it's definitely not the intended path. That's very likely to overwhelm you with AI response. 20 planets out of 80 is probably a good maximum, though you could get away with a bit more than that depending on the situation.
I can't figure out how to take a screenshot
Either printscreen to get your whole screen for the clipboard, or F12 to dump a shot of just the game to the RuntimeData/Screenshots/ directory in your AIW directory.
I now understand that in doing so I have alerted one core world for each AI
Previously combined with that AIP this would just be incredibly painful. It's still nasty but at least the AI's reinforcements stop getting more voluminous around AIP 250 or so on 7/7.
Unfortunatly I now have a mass carrier force of 10k ships (mostly mkV) amassing which will presumably attack at some point soon. I have tried forestalling this by various methods (since back when their numbers where as low as 2000) by can't seem to hurt the carriers in any significant way and also can't let them through to my world a few at a time as they're able to pass through FF's.
Am I screwed?
I'll let some of the community players chime in on this and the rest of your questions (they play through situations more often than I do) but the carriers can simply be nuked if it really comes down to it. But there are less drastic solutions, too.