How can I stop my console from restarting my single player game when my internet connection...

the game might be a internet conection only game

that's not possible to avoid in certain games, some games recuare internet connection to start, even if you want to play single modes

[quote user="Back Lot Basher"]Specifically, my question is, if I'm playing a game save from Cloud Storage, and my internet connection drops, will I be kicked from the game immediately?[/quote]

Of course a multiplayer game can't continue if the connection drops, but if you're playing a single-player game from a cloud save and you lose the Internet, you can keep playing. When you started the game, the console made sure your cloud save was
sync'd to the local cloud cache. (There could be some games that reset partially or totally if the connection drops even during a single-player game, but that would be true regardless of whether you're using Cloud Storage.)

Well, I am setting my sights on the single-player mode, so how do I get these prompts for buying the subscription off my back and stop them from interrupting my game?

Try restarting your game or your console, i have had this problem before so i tried disconnecting the internet from my console and restarting the internet connection and that helped.