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So am I understanding this correctly: anyone can build an app or site that integrates with the steem blockchain and pays out steem to users of that new app, based on various rules regarding creating, voting, commenting etc.? (I realize this is probably a pretty dumb question!)

I think it would be much easier and better to build the integration without the upvote and reward like steemit, and instead users should deposit steem or sbd to pay for the answer to their question

Yes, ultimately, you will be paying out to users of Steem (the blockchain). Checkout steemtools.com for the projects.