I've been playing Giant maps lately, YnAMEP mod, and sometimes I have a start where an AI civ is right next to me. So after founding our borders are touching. Is there anyway, other than building a stronger military, to keep the AI from declaring war almost immediately? I seem to remember in Civ4 that you could gift resources and keep the AI friendly, is there something similar in Civ5?

I like Civ5, but I do think there needs to be more positive diplomatic actions. Your best bet really is to build a strong military & keep up your city defenses. Barring that, check nearby barbarian camps for captured civilians and return them; that earns you a huge bonus that may offset the covets your lands, settling too close, etc., penalties if you are lucky. You can try a DoF early on before things get ugly, but it's probably not going to fly. I think that's about it. When two civs are right on top of each other, there's not a lot of hope for avoiding a war.

Yep. Virtually inevitable.

And they'll make it sound like your fault, the buggers. "You're aggressively expanding" -