I think that you need to compromise on something.
I run; swim; cycle and play hockey. I also do core work. I used to do free weights until I got hit by a truck and couldn't even lift a teacup.
I'd love to be a really good swimmer but I don't think I have what it takes despite all my training. If I stopped running then I might improve but I'd never win medals and I quite like my running medals.
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Hockey, I've only started playing this summer and only train once a week for an hr and a half. I can still run on the days I play hockey, except if I play a full game as I have discovered. The one thing about being able to run fast is that you get put on the wing.
I really think that you should work on your core to look good. Weight training is ok up to a point but only to a point. There are a group of lads that attend the gym that I do. I have watched them. The one that looks the best, has the most grace and poise when exercising is atually the least bulked up of them. He looks better and stronger. Maybe look at some youtube videos of urban gymnasys to see what I mean.
Personally, I would not run everyday. I would not visit the gym everyday. If by 'tank' you just mean that you want to feel strong in both your running and gym workouts then you also need to incorporate rest periods into your training. Going at it all guns blazing is just going to destroy you.
This year I felt the strongest after having a weeks holiday in which all I did was sea swim. No running and no cycling.
You're already doing quite a lot, as had been said, work out your priorities and what you really want. Also, I think that if you do take a week out of hard training you will see the benifits of rest and actually reap the rewards.