How to export client-entered content for a specific channel to keep my local database in sync with staging
There's currently not an easy or clean answer for this.

There are some great answers here on different solutions for handling data syncing/migrations across multiple environments.

We've got some ideas of our own on how we'd like to make this less painful for users (it's a PITA for us as well), with syncing / importing / exporting data between Craft installs, but unfortunately we don't have an ETA on it.