Zoom lens for 5d mk 3

100-400 is the classic choice although some aren't keen on the push-pull design.

I'd go with 300F4 L IS and a 1.4x extender.
 
Re sports
#1 What sports do you want to shoot?
#2 Where will you be shooting from (specator or media areas)?
#3 In what lightging (outdoors in daylight, indoors, aoutdoors at night)?

Re wildlife?
#4 Will you be shootging small wildlife (eg small birds) or wildlife in the wild. If you are then that really means 400mm minimum.
or will you be shooting larger wildlife/wildlife that is used to people then 300mm may be enough.
 
Sigma 150-500 OS is one of the best 'bang for buck' lenses there is. Mines on my 5D3 nearly all the time in the summer for wildlife, airshows etc. The Sigma 120-400 OS is also very good, very similar IQ to the 150-500, just a bit smaller.

Going up in the price range you've got the Sigma 50-500 OS and the Canon 100-400 L. I had a Canon 100-400 and the Sigma 150-500 and after much testing, with a 5D3 and a 60D that I had at the time, I decided the the Sigma edged it on overall performance and I sold the Canon.
 
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