Actually, the 70-300 is pretty good for sports ... outdoor sports, that is. If price is a concern, an alternative to the 55-300 and 55-200 is the Tamron 70-300 SP VC USD. The Tamron 70-200/2.8 would be better for indoor sports, but it's not very quick. The Sigma and Nikon are better choices, but they're much more expensive.
You never mentioned anything that would lead me to believe you needed the 10-24. If you've got something in mind, the Sigma 10-20/4-5.6 is about as good for a lot less. (It has a lot of fans around here.)
The Nikon 16-85 is an excellent walk-around lens, followed by the Sigma 17-70 and 17-50/2.8 (the latter could fill in for some of your low-light shooting as well. 24-70 lenses are more limiting on APS-C bodies.
- The lens is the thing.
- 'Full Frame' is the new 'Medium Format'.
- "One good test is worth a thousand expert opinions." - Tex Johnston, Boeing 707 test pilot.