I don't have mobile me or gmail, but I'll answer as much as I can.
For the notifications, no you can't clear them all. Well, if it's an rss reader then maybe, it just depends on the app I guess. In mail I THINK you can just move the "read mail that's actually unread" into a new folder. The badge should only reflect the amount of unread messages in the inbox itself. Facebook doesn't have a way to mark all as unread, so you'll have to deal with that badge showing up. Personally, I like it to stay there so I always know how behind I am in checking my messages. I used to consider my computer to be my main email too, now I tend to keep a mental list of things I need to do next time I log on from the computer that I couldn't do from my iPhone (either in the mail app or safari). I'm proud to say that I'm ALMOST desktop free when it comes to email
AIM, along with the other apps, will run even with the screen off for 30 min before the iPhone shuts it down. At least that's the conclusion I've come to, can anyone confirm/correct that? Once Apple releases their already promised push service then IMs and anything else the apps want will pop up just like mail and SMS even while they're off. Until then, yea, you're signed out when you exit.
I'll skip over your email questions to the autolock one. I use yahoo mail and push works perfectly about 98% of the time, and I've heard that mobile me does now as well so hopefully it's just a settings issue that someone here can help you with. I can't stand the autolock either because I use my phone at school and work a lot and end up taking more than 2 min to get back to it often enough to be annoyed when the screen's turned off again. I keep auto lock off and have gotten into the habit of hitting the power button when I'm done using it. You wouldn't want to be able to turn the screen on without the swipe, imagine the horror of pulling it out of your pocket and realizing you accidently turned it on a few hours ago and now it's opened up in some random mail folder. You can, however, leave the app open and the next time you unlock it'll still be there. The SMS app will open to your new text automatically if you unlock the screen before it blanks the screen after notifying you. The text in the "slide to unlock" field changes to "slide to reply" or something like that. The SMS app also opens to the newest unread message automatically when you tap the icon.