Something wrong with my Xbox controllers?

So I got some Xbox360 controllers. One of them I've had for years and the two others are brand new. For some reason, they've all been acting up recently. I can be playing a game or watching Youtube on Xbox Live and they just shut off. Without any reason and my Xbox doesn't even tell me anymore that it's off. I've checked and changed the batteries countless times, re-synced with the Xbox, given them a break. And I just don't know anymore. It's really annoying and sometimes when they shut off during a game, it's only during certain times like if I suddenly get attacked or there's a loud noise. And they won't turn back on for a while too. The buttons will flash then turn back off. What should I do about this problem.

My mom says it might be the Xbox and not the controllers themselves but I really don't know.

And another problem if it's linked to the first one, my xbox will randomly disconnect from Xbox Live. Once I connect it takes just 10 seconds to five minutes before loosing connection. I've reset the router and put the Wifi password in again but the internet is just wacked up. I thought this problem could be linked to the problems with the controllers but if someone knows, it would really help. The xbox is in good condition too. I only got it a year ago and never had this problem until just a few months ago.