just as a point of interest, most of the time i get iphone's with charge port problems, its just dirt stuck in the port that stops the charger going in properly. try plugging the charger in, if it doesn't sit flush against the bottom of the phone, its probably just fluff inside. try pulling it out , CAREFULLY with tweezers or something else that may fit in the charge port
sounds like a bad connection to the antenna if your having to press on it to recover signal. its possibly come loose when its been dropped. if its not that quite often restoring the phone via itunes fixes this
hi Could be a few things. Firstly the screen not fitting correctly could be a really bad screen, but from my experiences with these phones the metal housing (back part) is usually bent on one of the corners at the point of impact that smashed the old screen. The colour difference could again be a bad screen. it could also be damaged ribbons but also try cleaning the connections on the logic board and the screen with isopropyl alcohol.
Have you tried new batteries? The original batteries aren't great. Also try holding the home button down for up to 30 seconds. If your still stuck you can download an app from App Store or play store for your tablet/phone you can use. Just search for firestick remote
As Doots says, if the seller is not responding, chances are its a stolen or lost phone. You need the original owner to delete it off their icloud account. Unfortunately if you don't know who they are your stuck. If you paid with paypal, try opening a dispute. you may get a refund.