Re pumps on the skeet range - It depends on how serious you want to be. There is a lot going on with a pump gun on doubles. In competition skeet you need to break 395 or more out of 400 targets to win a shoot, more like 400/400 to win a major shoot. No one does this with a pump.
Sunday afternoons at the range, sure. Leagues - well, maybe. Our leagues are team events so you may be handicapping your squad by shooting a pump. If the squad is ok with it or the squad is a squad shooting pumps for a reason, why not. I shoot my grandfathers old fox sxs occasionally - it's not conducive to high scores, but it's fun.
People that shoot pumps, a lot, rack the slide without even thinking about it - if you shoot enough to get to that point it won't be a problem. Until then though, It's one more thing you have to consciously think about and it's one more variable you have to deal with in that second or so you have to make a shot.
As far as skeet guns go - the trend these days is for a 12 gauge o/u (so you can tube it for sub gauges), heavy, and 30" bbls. For an autoloader 26" or 28" bbls would be fine.