I can't promise you that it will work without the PRAM battery, but I know most desktops and laptops that I have messed with have booted without said battery. My opinion is you should be fine without it.
I can't promise this is the case, but it really does sound like the battery; especially if you take it off the charger and it "becomes a 7" brick". I know some Android devices require some battery power to start up, but I'm not sure why they do. It's probably your best option to replace that first.
You said you paid for it and expect it to be fixed, did you pay someone to repair it or did it arrive dead? Do you have the option of returning it? That'd be the best way to go. Otherwise, leave the charger in for a couple hours, preferably plugged into a certified Apple wall charger. If that doesn't revive it, and you're comfortable taking things apart I'd recommend you check the water damage indicator. If you need help with that then let me know.
Definitely try it with the cable as well; you can rarely assume the battery has a charge. Found this at "howtogeek.com:" "Tablet Won’t Power On or Charge If your Nexus 7 appears completely frozen and won’t power on or even charge, the most obvious solution is to pull the battery and reinsert it. However, the Nexus 7 doesn’t have a user-serviceable battery. Instead of pulling the battery, hold down the power button for 30 seconds straight. You may have tried holding down the power button already – but you won’t see any results until you hold it down for the entire 30 seconds. This process fixed a Nexus 7 that refused to turn on or charge."
I would check your battery connection and make sure you didn't break or tear any connections or cables. I had a friend whose iPhone wouldn't charge and kept shutting off randomly because the battery wasn't properly connected. And of course, here is the guide if you need help. iPad 2 GSM Battery Replacement
It sounds like you should resolder the metal connections on the logic board. Just disassemble your external case and find the logic board that plugs into the hard drive. Then look for broken connections that you can resolder. It should be relatively simple, though in just a few minutes of searching I wasn't able to come up with anything super helpful as far as guides go.
This should help you. https://www.youtube.com/watch?v=w-4M51ypamk If you need help finding the part, here you are: http://www.autozone.com/autozone/parts/S...-Turn-Signal-Light-Bulb/2000-Chrysler-Town-Country-AWD/_/N-jeoe5Z9cjcs?itemIdentifier=89682_0_18440... The part isn't guaranteed to be the right one, since I don't know the specific edition of your vehicle, but it should be close enough. Best of luck.
Can you just replace his power button and use that to turn it off? I've done that with my iPod Touch 4th gen, and it works just fine (I could have been lucky). Of course, if you don't have the parts then that's not an option. If you don't have the parts could you pull one off of a working/broken extra? Without a power button physically attached I don't know of any ways to turn it off though, sorry.