There is no lack in APIs honestly but If you're planning to provide something better when it comes to request limits, features, and competitive rates then people will definitely use it.
Definitely there is! Most of the existing API's revolve around sending/receiving bitcoins, checking balances, trading and other stuff. I don't recollect seeing an API which helps implement a bitcoin mixer and is used by many. Private services don't make their API's accessible to everyone. I'm sure you will find a lot of similar projects which needs a public API.
Bitcoin mixer is pretty easy to make but the problem is capital. Don't you need both trust and funds which both I don't have in order to start a bitcoin mixer?