The cutters are made out of ABS Plastic, which is the same plastic as LEGOs. It’s very sturdy. For cleaning, I tend to go the lazy route and just dump them all in a bowl filled with water (usually whatever bowl I had the flour or dough in, because I fill that with water, anyway, when I’m done) and just let them sit for a while. After they’ve soaked, I just throw them right in the dish washer on a normal cycle, and it almost always results in totally clean cutters. Admittedly, I’ve got a good dishwasher, so it’s possible that you might need to go into the cutters with a toothpick or a brush, to get dough out of any small corners and crevices.
If you’re the type to clean all your tools the second you’re done using them, and not leave things soaking in the sink, then you’ll probably need to go over the cutters with a brush or toothpick to dig the dough out. But nothing that you wouldn’t have to do with any cookie cutter.
As for the dish washer, I’ve always used the normal cycle, not the heavy-duty one, but I’ve never had any problems at all. No indication of warping, even on a heated wash (It’s one of the reasons I don’t use PLA plastic instead – it warped in the dishwasher when I tested it.)