Sling user here.
I may take as you train so shall you fight to the extreme, but I train with a sling, especially using it for transitions.
The sling I use is cinched up pretty tight. It's a simple matter to scoop the sling under the forend hand, and there's really nothing to catch it on in my house, it certainly isn't hanging low enough to catch on doorknobs.
BTW, I WILL NOT ATTEMPT to clear my house. Right now, it's me and my wife. Our room is the safe room, so it's basically a one sided barricade situation, with the sling being a non-factor.