If you resign without any moves played, it is counted as an abort with no rating change.
Can a game be removed from the record by mutual agreement?
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One move was played 
I was clicking everything I could see to terminate it.
Don't worry about it. Over time here we all win games and lose them, resign and abort them etc. all for a million reasons. Give it a few days and quite a few games, it won't feel like it matters if a weird game is on your record. And, aborting games will only hurt your fair play record If you do it habitually.
"Daily" is like correspondence chess and will only time out after 1 day or more. "Live Chess" is where you'll find 10 minute games - go to "Play" on the left, then "Live Chess" then make sure it says 10 minute at the top right.
Yeah, you're right. Resigning on move 1 to an 800 level player probably only knocked my rating down 400 points or so. But rating's not everything, right? 400 here, 400 there, no big deal. It's not like anyone pays attention to ratings in chess, or use it as a preliminary yardstick to gauge credibility, huh 
(Yes, I understand that over many games it will largely wash out)
I'm very familiar with postal chess and understand that Daily simulates that. Evidently I landed in Daily because of a frustrating attempt to simply start a game with a friend, and suffering counterintuitive site responses to clicks evidently eventually led me to try Daily just to get a game started (with the intent to then immediate change the clock setting). I really wish I could reconstruct the counterintuitive sequence that was so frustrating; if I can piece it together, I'll report the problem more formally, and you'll understand why I was starting to experimentally click on buttons. I suspect it was one of either clicking Connect, clicking play, hovering over play and clicking Live Chess, or clicking on Challenge somewhere, or clicking New Game. (It seemed like no matter what I tried it was putting me in the "want to play somebody queue" instead of "start playing with a friend") The thing is, for everyone still participating in the site, if they suffered the same thing originally, they no doubt found a click sequence that works, and long forgot the counterintuitive click sequence; and others might have given up, and they're long gone.
@ EddieHaskell,
2 things...
1. Dropping those ratings points is no big deal. If you dropped 400 points by accident in your first game, and you're truly at or above the level you started, your rating will be back up in no time - mostly because you'll be playing opponents who are no where near as good as you, despite being rated the same as you. If you're 400 points down from your real ability, you should enjoy a good string of wins for a while.
2. Who cares what other people think about your rating? If they feel you aren't credible because of your rating, that's on them.
My first game here ended up in "daily" or postal mode and it didn't seem to have any possibility of timing out so the only recourse that I found was to resign the game. I hoped that it would just time out due to nonresponse but it appeared to have a 24 hour clock on that.
It's really frustrating to try and start a game with a specific player because IIRC Play/live chess/ Play a Friend will not lead to being able to play a 10 minute game (it seems that there were only 1 or two choices and one was daily?}. I'm so sorry I didn't record the click sequence to show that here. It seems that the only way to get the game started was to go through Connect and click on Challenge, or something like that. At any rate, since it wasn't working intuitively (for example, I seemed to place myself in the queue for anyone to challenge me, though I just wanted to play a friend), I just started clicking manicly and perhaps I simply deserve what I got!