Should I sell my iPhone 5 with Flappy Bird on eBay/Craigslist?

Won't my Apple ID also go along with the phone if I sell it? I'm just wondering about the possible consequences if I delete everything except the app, Flappy Bird. Won't the person who receives my phone have my personal information from my Apple ID? Basically, how can I sell my phone with Flappy Bird and have complete confidence that the person who buys it won't be able to access any of my information? After all, it would be impossible to do a compete format of the phone or else I'd lose the game. Thank you for considering my question and any answer would be greatly appreciated.

If I buy this iPhone on amazon can I use it in Peru?

Well I'm thinking on buying an iPhone 4 and I saw this ones on Amazon: http://www.amazon.com/...004ZLV5UE/
I was wondering, if I order them and unlock them, will I be able to use it here in Peru? Or does it have a mobile plan already or something? I'm not familiarized with this kind of things so please be as brief as possible.

Added (1). What I meant is: Can I unlock it and bring it here to Peru and use it with a mobile plan from a company here in Peru? I'm adding this information because I think my question wasn't clear enough.

Sim card stuck in iPhone 5 without tray?

Sim card wasn't going in the tray properly so I thought that I could put it in then put the tray in but now its stuck, I've tried the sim card opener thing and tweezers are too big.

How to change an iPhone's imei number?

I recently got a iphone 4 from my aunt and I want to upgrade from my regular phone to a smartphone. The problem is that I do not want to pay for the ridiculous data plan that at&t forces smartphone user's to pay whether they want cellular data or not. I believe it is wrong they make me pay for a service I don't want in the first place. So I'm trying to hide my phone from them and trick them into thinking I'm still using a general phone. I have heard that they can detect if you began using one by the imei number and begin charging you. Is there any way to change this number?