Yes. The money visa gives away has to come from somewhere. Same as with your coin btw but lets stay with the visa example. Where could visa get the money from?
- increase tx cost
- lower profit
There is a good chance they wouldn't have to increase the tx costs, or lower their profits. In the situation where they actually started doing this, people might use their creditcard more often and using the extra profit this generates, Visa would be able to pay out prizes. They don't gain anything by doing it, but they wouldn't make a loss out of it either.
But your main point is 'the money has to come from somewhere', and for a crypto-currency that's not really relevant, unless you can explain where Bitcoin's 21 million coins come from?
Miners will fill block with their own transactions (transferring bitcoins from left pocket to right) to increase their "profits" (chance to win reward).
Ofcourse, this must be prevented. I already suggested a couple of fixes for this (like requiring them to broadcast transactions to the network before including them), but if it turns out no solution for this is possible, then this coin cannot exist. But even in that case, I'm still curious about the philosophical/economical implications of such a network, since I wasn't planning on actually developing it anyway.