Since they're both international flights, you will arrive into and depart from terminal D.

Your best bet to store the bag might be the hotel bell stand at the Hyatt in terminal D. Most hotels will check a bag without question, and if there's any resistance, I'd bet it goes away with a $10 tip.

As far as relaxing...
$35 will get you into the Club at DFW.

If you don't want to spend anything, the best places to relax at DFW are the Samsung Travel Lounges. These are small, semi-open rooms, free to the public but underutilized, thus quiet. You'll find chairs that are much cushier than the ones at the gates, plus desks with electric outlets to work on your computer (wifi is via t-mobile for a fee). None in terminal D (they're at gates A29, B8, B28, C8, C27, E8, E31), but you have lots of time to to get there. The air-side Skylink will get you to any terminal in 10 minutes or less.

If your flight from Canada has customs pre-clearance (most do), then you'll arrive airside. But if you have a checked bag, you'll go landside to claim it, and if you have to wait to check in for the next flight, you won't be able to get back airside until you do. And there are almost no amenities at all landside in the DFW terminals. My advice there would again be to head to the Hyatt hotel, to the restaurant, bar, or even just the lobby.