Getting closer is almost always the best option. I think any of the lenses will give you great results, you'll need to practice with the setup some and get your routine going. Good shooting technique counts for a lot.
Generally, IS does not work beyond 1/800 - 1/1000th or so, depending on the components. Some people claim they get worse shots with IS on beyond 1/1000th some say it's OK and the lens figures out to turn it off. I think it varies between lenses. Just like at slower speeds some people say turn it off on a tripod, other say no leave it on if there's wind vibration, etc. Anyway, for faster shutter speeds, you don't need it and likely don't want it turned on (all the lenses have a switch on them for on/off and the VR mode: normal, sports).
All Nikon F-mount systems only have IS in the lens, if they have it. The Z-mount has in-body IBIS. Rumor mills say the D6 will have IBIS, but that would be the first for Nikon in the F-mount.
All the lenses we've talked about have it. Nikon lenses call it VR (vibration reduction). Sigma calls it OS (optical stabilization). Sony calls it OSS (optical steady shot). Tamron calls it VC (vibration control). I hope I got all those right; it is an alphabet soup