Phone rebooting after battery replacement

I highly suspect your problem is pry damage just to the right of the battery connector. You've likely knocked off one or multiple components with your spudger, fingernail, tweezers, or whatever you used to disconnect the battery. This is a common mistake, but requires micro soldering and likely a parts board to repair. Personally, I'm too blind to do it without a microscope. It's advisable to use your heat gun to place these rather than trying to get on it with an iron. The folks with the nice equipment use hot tweezers. I'm not nice and I don't have hot tweezers. Hakko makes a nice set of hot tweezers that I'm super envious of.

Google "battery pry damage iphone 5" to find some pictures of the damage I'm talking about, its too late in the evening and I'm too intoxicated to do it for you.

Best of luck with it friend, to better see what you're dealing with, without buying a microscope, Radioshack sells a little cell phone camera adapter microscope that does a decent job for observation and finding these things (can't work under it of course) tell the person at radioshack you want a 630-1331, surely they can deal with that (its their own sku number) and hopefully find you one.