Yes, you definitely have pry damage, but possibly other things going on as well. Let's deal with the issues separately:
1>>Pry damage. Your pics are fuzzy, but I can clearly see that you are missing at least U3 and DZ101_RF. I am hoping that your six U3 solder pads are ok, they look a bit mangled in your photos. Having them bridged together like that may be the cause of your iTunes restore error. If they can be laid back down nicely without tearing off (don't try it--requires heat!) then this phone is a fine candidate for our guaranteed pry damage repair. I will send you a link to my pry damage service via email if you want to get this done. If I have to build microjumpers to replace torn pads it can be done if you are missing one or two pads, and once I did one missing three pads---pad repair is more difficult than just placing a new component.
2.>>Restore error. Since you have an obvious hardware fault, a restore error is not surprising, however, I have seen two or three times where pry damage was coincidental with 'screwdriver slippage' on the nearby glass chip just below the pry damage area. If this chip is damaged, then we are not able to replace that chip--but maybe someone else can? You might want to look closely at it before sending the phone in.
3.>>No boot from battery. In common pry damage, the phone can still detect the battery. A phone that can not remain on without usb charger is unable to detect its battery and that is not a symptom of classic pry damage. This is a separate issue. Unfortunately, with multiple problems going on, I think that you might first have to get your pry damage straightened out in order to be able to troubleshoot the no battery problem. That one could be as simple as defective battery, or accidental loss of tiny little components that are near the dock connector that are required for battery function, or damage to the connector itself.
I will send you a reply to your email, and hopefully we can get this phone straightened out for you.