I shoot a lot of action. I also own a 5D3 and a 1D4. The 5D3 is a very competent camera but for shooting action the 1D4 operates in a different galaxy. No comparison. The 5D3 is not a sports action camera. The AF is fine, but frame rate, weaker ergonomics, less solid weather proofing, shutter life, and very importantly buffer depth are the factors that make the 5D3 a non-starter when compared to the 1D4. It does tick a lot of boxes, but sports action was never part of the 5D3 job description.
If your primary objective is a hot sports action camera and you're choosing between 5D3 and 1D4, don't even think about the 5D3. If the budget is there, stretch to the 1DX and be amazed...
I have a 5d3 and 1dx (and have tried a 1d4) PW is spot on, the 5d3 is a capable sport camera, but for serious sports photography the 1d4 and 1dx are much better. I rarely use my 5d3 for sports now, only as a second body to the 1dx.
also...fyi - the 1DX was on the canon usa refubished website for sale at $5500 a few weeks ago...it's still listed, just out of stock
and 300 vs 400....decide whether you like using a monopod or not...with the 400, it's a must, but I can shoot a whole game with the 300 attached to a black rapid strap