If by "restart", you mean start a completely new game, then no, you shouldn't need to do that.
As for the interactions...
When you buy a game disc these days, pretty much the entire game is installed on your console's hard drive. The disc means that there's less to download before you start playing the game (because a lot of it is copied off of the disc). And you have the option to pass the game along to someone else later on so that they can play it instead of you. The upshot of this is that the DLC expansion doesn't really care whether you're running the game with the disc, or downloaded it onto your console. The DLC will work just fine either way.
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