House edge affects the bet payout. If the house edge is 1%, the payout would be 1.98x, which in the long run would make the house win by an average 1% of the wagered amount. If you don't bet, the only fees charged is during withdrawal if it does charge a fee during deposit, you should stop using that casino. That being said, use gambling sites aren't always legal in many jurisdiction. They do keep logs too.
I know a few mixing sites. Their fees are like insane. I have seen one of the sites charging as much as 20% as fee. Compared to that the house advantage given for the gambling sites are quite low. Even if you have to wash it three times, on average you are going to lose only around 3% of your total coins. That said, the transaction logs are an area of concern.
You definitely didn't get my point. The house edge does not affect the balance till you decide to bet it. There will be no fees except the withdrawal fees which is minuscule. That being said, mixers does offer more assurance than gambling site. Bitmixer is one of the lower costs ones.