1) You can use the Nikon FX lenses on your DX camera (D3200) as long as they're AF-S, otherwise you'll be manually focusing, which will be a challenge in sports photography.
The DX is a crop-sensor, so you'll be getting the effective field of view of a lens 1.5x as long, so it actually appears to be getting closer than it would on an FX body. Most of the AMAZING sports pics don't come from the "other" end of the field. You'll notice folks set up in the endzone, on the baseline, etc. and focus on the action coming toward them. If your team is going the other direction, you'll need to be on the other end. Period. Otherwise, your 800mm will be getting a nice zoom in on their backsides anyway... (FWIW, an FX lens will work on either a DX or FX body. A DX lens won't work on an FX body without some forced vignette around the edges. If you plan on EVER moving up to an FX body, buy FX glass to reduce the cost later.)
2) Use what ya got and see how you like it. That 55-300 should give you an effective FOV of 450mm at the long end. See if you can get some decent pics with that, crop to what ya want, and go from there. If you're satisfied with what you're getting, start to invest! I'd suggest shooting in bursts, as it's practically impossible to
time the perfect action shot. If you like what you're getting, you'll definitely want to update to a camera body that has a bit faster burst and bigger buffer, and perhaps more cross-type focus points, like a D7100 - 6fps (or D4s - 11fps, woot!). FAST SD cards (like 90 mb/s) are a must!
3) You'll want some faster lenses if you're shooting anything at dusk or night. i.e. 70-200 f2.8 VRII or something similar, and that gets costly. Start planning for that!
4) GOOD LUCK!!!