Yes. Sometimes Hotels have cheaper rates on Airbnb. Not sure why. Possibly because they don't want to advertise those lower rates on their own websites or they have exclusive deals with Airbnb.
Edit: Another reason is that Airbnb simply provides more eyeballs for hotels. Sites like Booking.com is very crowded already.
If you can, I would always call the hotel and ask for price matching. 9/10 they say yes and sometimes I even received a free room upgrade because I paid with them directly and not a 3rd party.
Yes, any hotel manager would be a fool to give 15% to 20% commission to Expedia/Booking/Airbnb rather than split that with the hotel guest and have both parties come out ahead.
I've had instances where I've extended my stay, and they've found me a cheaper room. However, calling the hotel itself prior to booking has always resulted in higher or identical prices.
It's anecdotal, but I've not personally found it to be worth the extra hassle.
Edit: Another reason is that Airbnb simply provides more eyeballs for hotels. Sites like Booking.com is very crowded already.