Hi Danzig, I have only previously used a 300mm mirror for a film camera but the principles are the same.
Pros, smaller and more lightweight than non mirror equivalent, less room needed in the gadget bag. Sony 500mm is autofocus. Not as obstrusive as a non mirror so more candid.
Cons, fixed apature, usually f8+ unless you add small supplied filters to increase. As a result usually a darker image in the viewfinder. Unlike the brokeh of a normal lens the brokeh of a mirror lens will record highlights as 'doughnuts' as a result of the mirror therefore it's a 'love them or hate them' thing.
Don't be tempted by the cheap 'T' mount mirrors of Samyang(?) etc which get a mediocre feedback at best.
Mirrors have their place but in limited uses, and I've never seen a sports photographer with one. Maybe you could check out the Sigma 200(?) to 500mm 'bigma' zoom which is apparently highly regarded?
Best of luck!
Don't you just love helping other folk spend their money

Phil