When you start matchmaking does it give you a notification regarding NAT type? If your NAT isn't open you can sometimes have difficulties connecting to games.
If a map doesn't load and the map changes it means someone in the game couldn't load that map, if you keep getting kicked out of games and the maps keep changing I'd put my money on it being your copy of Halo. If you're using downloaded maps try playing the Mythic playlist and see if the same thing happens.
The final thing I can think of is your disc drive is coming to the end of it's life, you can buy a disc drive cleaning disc for around £5 ($8-10), put that in your Xbox, let it play it's jonty tune as most of them usually have one and then bobs your uncle your disc tray is good as new.