How does a game like Satoshi Dice avoid de-anonymizing users?

When considering games like satoshi dice or many other more typical online casino games, how do they handle awarding prizes? It seems like they pay back to the original input that sourced capital to them yet I have heard that addresses should never be reused for receiving payments.

My Question then is two fold:

1. Do gambling services that pay back to inputs that have paid them create a de-anonymization risk for the network?

2. If there are multiple inputs how do these gambling services decide WHICH of those inputs to pay back to? Is there a "Safer way"/"Preferred way" to select such an input?