1) Even if 99% of people use mixers with good intentions to just have some reasonable privacy, you never know what was sent there. What if you pay someone after mixing these funds, this someone goes into an exchange, deposits the coins and the exchange is triggered due some flagged coins? This person will then point at you as the source of the funds and now you are in trouble. How do you avoid this? Btw this includes privacy wallets, not only mixing sites, I've seen people report about this.
Isn't this a non-issue? You are responsible for the transaction, and nothing further than that. Flagged coins are stupid, and I personally couldn't care less about it. The transaction would concern both parties holding up their end of the deal and nothing further, unless you are supposed to be transacting in "untainted" coins. You shouldn't be responsible for whatever happens outside of the deal, and furthermore you are not in trouble for using a mixer because you can prove that you are not involved in any illicit activities.
2) In order to send the coins into the Android software, you have to transact from Bitcoin Core, into the mixer, then into the Android software address. This is a lot wasted in fees. There is no way to save some money in this process?
Then you would have tradeoffs. If you don't want to incur additional fees, cut out that transaction to transfer to your Android wallet. In the event that your phone gets stolen or your phone gets hacked, then you would lose more than you are supposed to. Unfortunately, you can't have the cake and eat it too.