You can, but don't need to and probably shouldn't unless you're pedaling home a breakdown.
To the point of bikers ignoring laws, most of the laws are designed around cars. You don't need stopsigns and lights if everyones on bikes, you can totally get around safely and at actually lower impact to cars if you're not stuck following car traffic rules. I don't mean forgo all safety and blast through intersections, but if it's safe to go through a red on a bicycle it's actually improving traffic flow and probably your safety on a bike also... The downside is it upsets drivers, who in a car you wind up pretty conditioned to view everyone outside the vehicle not as people but as obstacles to getting where you're going. And you wind up feeling wronged if someone gets to go when you have to wait. Seems unfair. Sadly they also don't support better and more functional alternatives like bike lanes to improve both driving and cycling.
TLDR I think it's important to respect the intended purpose of those bike lanes which is providing safe efficient routes for people biking, and not to be a convenient buffer for slower vehicles like your moped. Walk on the sidewalk, bike in the bike lane, ride/drive on the road. It's safest and most effective when you're doing what you're supposed to where you're intended to else you confuse and obstruct.