Navigating Boston by car is challenging at best with an adult co-pilot (for example when you end up in a tunnel, your phone/GPS will not work well). Plus you'll be tired and the traffic will be terrible! I also can't picture somewhere that I'd want to pull over with kids to nap in the airport area. It's not that the area is dangerous, it's just very densely populated and I would feel like I'd wake up to someone looking in my window.
That said, here's what I would do. Without knowing what exactly neighborhood you're staying in, I'm assuming that you'll be reasonably close to public transit. I would use public transit or Lyft/Uber to get from Logan to the Airbnb and ditch the luggage there (if that's possible). Not sure if it's an option, but you might consider booking the Airbnb a day earlier so that you can get in as soon as you land and get in a nap.
If you can't store your luggage at your Airbnb in the morning, you could start out with sightseeing. The best way to do this is by taking a water taxi from the airport to downtown - buy a package that includes a day's worth of luggage storage (
https://www.roweswharfwatertransport...uggage-storage). Once you've shed your bags, you can find a cafe or park to sit and rest. There's a lot of options along the waterfront.
Once you've checked into your Airbnb, I would use public transit to hit the Freedom Trail and USS Constitution over the next couple days and not rent a car until you need to get to Plymouth and Montreal. If you can't find good dinner options that are walkable from your Airbnb, you can always use Grubhub/Seamless to have food delivered. When it comes time to rent a car, there are a number of non-airport car rental options and they're typically cheaper than the airport.
Boston is awesome! I'm sure you'll have fun!