Did you give
Semi-Restore a shot? I recall iLEX rat lets you keep apps and their data and only re-initialize Cydia, make a full back up and fire up Semi-Restore, do not restore from back up before trying to launch any of the stock apps. Although both iLEX and Semi-Restore at first glance do the same thing but there are leftovers that are left behind by iLEX that Semi-restore takes care of.
Your device is yours to use and only you can decide how to use it but i can't stress enough how notorious appsync is when it comes to causing that particular issue, if you ever installed it try to get it from a different repo because removing it doesn't always fix the problem especially with tons of mods put online on multiple repos, remove them all and search for the most stable version out there since the damage already done i guess you could just install a working version at least.