Temperature changes don't affect cold/flu incidence ... so arenas per se don't create the problem. What does is having lots of people around, particularly kids. So, when LTS kicks in in fall and winter, having little ones around sneezing, etc, spreads germs ... particularly as they may not be as capable of hygenic actions. Plus, more people are physically inside the arenas as winter sports ramp up ...
Then, you get the back-to-school germ pool issue when kids are shoved back in to classes and they spread germs around and around.
If you are finding that your coughs and colds are worsened at the arena, then, that's a bronchial issue not cold and flu.
With comps, you travel, you meet a different germ pool ... you don't have resistance to the new germs. You get sick. If you are shifting time zones, then, that reduces the body's resistance to germs too ... and if you are flying, well, you're trapped in that airplane with all of the people and germs ...
Yeah, my kids swam competitively, play baseball, dance, etc etc etc. If anything, I think we had more disease type stuff with pools, as they humid and muggy air in change rooms and around the pool were great breeding grounds for germs ...