Wider tires on the Walmart bike is a reasonable idea, but expensive Compass tires are overkill. They're wasted on that bike. Get something fat and with some flat protection. Compass tires are not only wonderfully supple and fast, they're also delicate and wear faster.

Nah, these days 25mm is basically the standard size on race bikes. For Paris-Roubaix they'll go to 27-28 or sometimes even 30mm depending on the team, tire sponsor, rider preference, etc.