I'd say 90% of people use normal alpine skis, the only difference is weight and in my opinion, it's negligible. As far as boots, a/t boots are lighter, flex more in walk mode (for a more natural stride) and have vibram soles for climbing around on rock and such. The trade off is softer flex, less power to drive the ski. I've only used alpine boots but would love to have some a/t boots for long tours. I don't think that the alpine boots have held me back but i'm sure it would be easier with a/ts.
The most popular a/t binding is the fritschi freeride and it does quite well. I know plenty of people who use this as there only binding day in and day out at the resort. There is some noticeable slop on hard groomers (for me anyways) but i've been told you get used to it. Users opinions are going to be varied because straightlining chutes to large cliff drops onto hardpack are normal for some people and the freerides won't last forever doing that. But for the average skier, freerides are burly enough.
As far as the prophets, if you can ski them well in almost all conditions inbounds, then you should'nt have a problem in the bc. As far as skinning with them, just make sure you get skins that cover at least the whole waist (minus edges), and preferably cut to fit the whole shovel and tail also. The wider the ski with full coverage skins = more grip for climbing steep hills.
Just remember, fat alpine skis and thier big skins and alpine boots are going to be heavier then a deicated a/t rig.