Mobile Passport

Yes, you still need to go through customs even if you have Mobile Passport. The app lets you send your info a little bit ahead of time to speed up the process, but you will still have to present your official passport and briefly talk to an agent. Also, your bags may be checked at random, regardless of whether or not you have the app. You will be able to use a designated Mobile Passport line that should let you bypass the majority of the crowd.

Currently there are 24 airports and one seaport that accept Mobile Passport:

Yes you will still need to present your official passport to U.S. Customs, not to mention the officials of whatever country you travel to. The passport info that you have entered into the app allows you to send it a little bit ahead of time (basically once you land and have Wi-Fi), thereby speeding up the process. A custom agent will still have to verify your info by checking your passport.

TSA Pre-Check is designed to speed up getting through security upon your arrival to the airport, while the purpose of Mobile Passport is to facilitate a quicker customs experience upon returning to the U.S. Mobile Passport is not involved with any pre-flight procedures, so the two services are basically unrelated. There is, however, a link to enroll in TSA Pre-Check as soon as you open the Mobile Passport app.