Experiencing multiple aberrant connections to my IPAD leaving me with evidence that my device is not secured, no matter if it's replaced or erased.

Question

Several days after I got my IPAD my home network was physically accessed by neighbors who put a splitter onto my cable and routed my traffic into their apartment. They also had cable service of their own, so it wasn't to use my signal.  They have however used my subscription video services through Amazon and Netflix, and, if I purchase a film through Amazon Instant download, they are the first to download it onto their hardware causing me to have to call Amazon to re authorise me to download it.  I live alone so no one else has my log in info. 
It doesn't stop there! My Facebook account security settings show me I am usually connecting through a city I don't live in, on an operating system I don't own. To see these settings go into you security settings in Facebook and its on of the last choices in the list. At the moment I have a brand new IPAD and have never logged into Facebook on it.  My Facebook app shown in IPAD Settings, is secured indicating no Facebook account has used the app, yet a networking app called IT TOOLS--a Swiss Army knife of networking tools shows that a Facebook IP address is connected at this moment, and again, no Facebook is engaged through their app, or in Safari.  
Today while composing a note, I noticed several notes on the left side of the screen were being deleted as I watched.  I opened IT TOOLS to see what IP addresses were connecting and found a cloud based service and sent a report to their ABUSE@ email address, but I've done this dozens of times already.  Amazon gets a screen shot of the IP connections made that contains a date stamp from a singular point in time as their are no running firewall logs for IPAD. 
When I had my ISP come to see why my speed was 1/3 of what it should be, they reviewed Ookla speed test data and noticed that one entry was for a 3G platform, but this is a wifi only IPAD, how could an APP on my IPAD scan the connection speeds for my cell phone, unless my cell phone was set up to run as a hot spot, which I wouldn't do as they had a history of being hacked into by--get this--two women in their sixties who learned how to embed malware into Jacqui Lawson ecards, and once their unknowing recipent opened it, out comes a Trojan giving these sociopaths access to your computer.  One of these women was launching these attacks from a Baltimore convent she volunteered at, so not to implicate herself by using her own computer. 
Another issue involves screens changing, where I'll see my wallpaper change to one I never installed, then it seems to switch back to my work screen.  Associated with this is an abundance of IPV6 connections which leads me to what I believe is going on  a SLAAK ATTACK, where an IPV6 network is overlaid onto an IPV4 router creating a man-in-the-middle attack so everything I send or receive to the internet is caught by a computer that has clone software. I doubt these women are able to conduct such an attack, but I do know they have relations who are more than qualified.
Another thing I see is every time I bookmark something on Safari, it syncs to something else, but I never made a bridge allowing it to connect to anything else.
I'll be forever in you debt if you can tell me where to look on IPAD for info.  I was about to post my routing table because it's filled with 127.0.01 configurations and a few unknown mac addresses with IPV6 addresses, but realised not a good thing to show the entire internet community.  Thanks!

Answer 1

That's one of the weird things.. it recognizes it maybe 10% of the time. And usually, only after I do the two-button reset. Problem is.. since it won't charge above 2%, anytime I try to do a restore or anything like that using iTunes, my device shuts off and I lose whatever progress I'd made.
So, an update... after reading through a bunch of similar complaints (there are literally 1000's of them so there's NO WAY this isn't somehow ios7 related, thanks a lot APPLE ) I decided to try a restore in recovery mode. After 3 hours and several disconnections... I ended up having to just set it up as a new iPad, as the restore did nothing. Weirdly though... as I was doing the restore in recovery mode.. I noticed I'd gotten up to a 10% charge.. higher than it's been since September, so after setting it up as a new device, I turned it off and plugged it in using the wall charger. 2 hours later and I was up to 38%. Still not great, as my iPad, before ios7 could've fully charged twice in the amount of time it took for me to now get 28% more of a charge. And that's with a fully cleaned out device.. so that really ***** and I'm now more confused than ever.
But I'm gonna leave it overnight charging and see what I come up with tomorrow. Sadly, when I paid $600 for it in February, I never expected to have to play "wait and see" with it...