A teleconverter probably won't work with a T3 unless the lens is f4 or faster as the teleconverter increases the minimum f stop. A 1.4 teleconverter will increase the minimum f stop of a f4.0 lens to f5.6. Most Canon cameras require minimum f stops of f 5.6 or less to autofocus.
So, for instance, if you put a teleconverter on an f 5.6 lens, that would increase the minimum f stop beyond what Canon needs to autofocus.
[doublepost=1473333469][/doublepost]I have heard good things about the Sigma 150-600, but bear in mind it is big and heavy. Also, at 600 it is hard to track movement, and it is relatively slow.
If you have the money, look for a good used version of the Canon 100-400 (the first edition). There should be a lot on the market now that the second version has come out.
The Canon 70-300 is not a bad choice either. It is much smaller, lighter and much cheaper than the 100-400, but not nearly as sharp and doesn't focus as quickly (nor does it have the reach). You get what you pay for.
The 400 prime is a great lens for nature, but I don't think a prime is a good choice for sports.