You can be in a party while playing split-screen, but unless the second player has an Xbox Live account, they won't be able to participate in any of the voice chat options that Xbox Live offers. You can plug a headset into their controller, but they won't hear the party/game chat audio, and their mic won't function.
If the options to join a party aren't on the same buttons in-game when you're playing split screen, you can always use the Xbox Guide button to access the party menu and start/join a party from that interface.
Note also that you have to have a second controller plugged in/turned on in order for the "Press A to Join" message to appear in most games (MW3 included). There's nothing in MW3 or Xbox Live that prevents you from doing both at the same time.
As far as I know, there's no direct way of getting this information. However, you can safely go ahead and make your purchase anyway. Since downloads are associated with your account, they'll appear in the Active Downloads section of the Marketplace pane on the Xbox button menu no matter which Xbox 360 you log in from.
If the current Xbox 360 is not considered your primary console, the item in the queue might indicate that it's "Assigned to other console." If this happens, selecting the item and pressing A should bring up an option to download all items. Choosing to do so should update your account to consider the current system as your primary Xbox 360, and downloads will happen automatically when you log in from that point forward.
Microsoft has a Knowledgebase Article on this topic that might also contain relevant information, and Major Nelson did a blog post when this first became a feature.
I'm not sure if it's necessary to do it manually if you go through the above step, but you may also need to transfer licenses for any content you had originally downloaded on your other console.
To perform a license transfer, on your console, go to the Settings pane. From there, click on Account, and select the License Transfer option. As you can only perform this action every four months, it'll ask you to confirm. Doing so should transfer all of your existing licenses over to your new Xbox 360.
Best Answer
You can't sign in while a guest is also logged in. To get this to work you need to
Xbox Live won't let you sign in to Call of Duty Multiplayer with any profiles that aren't gold logged in. The "Guest" account technically isn't gold, so you're getting rejected for that. Adding a guest player is done on the Call of Duty server, not your Xbox.