Mixers tend to charge a flat percentage fee, but there may be an additional charge for them to send your money back to you due to BTC transaction fees.  There's also the fee of sending to the mixer as well, but from LocalBitcoins I think that should be relatively low.

One potential solution is to use Chipmixer, since there is no fee and you receive the private keys to your coins (so you can decide to set a low fee when sending your coins elsewhere).
Depends how much you value your privacy. 

It's possible that an entity would be able to attempt blockchain analysis to link together different addresses and wallets that you own, but it's not likely unless you either have a large amount of money; are buying something illegal; or someone has a personal vendetta against you.