Howdy! I'm a web developer based in the Washington, D.C. area with over 10 years experience in developing MySQL + PHP web applications. I've also been working with the phpBB platform since 2004, and you can see some of my previous projects here:
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However, this forum sounds like it's the most similar to what you're looking for:
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As with your project, RotoAsk is integrated seamlessly into the rest of the site, which is Wordpress. Users browsing the site often have no idea that they're using two completely different platforms.
Looking at the job description, I can install a basic phpBB3 forum with you removing the features you don't need, and also add in a few modifications to enhance overall security (such as anti-spam protections). However, the single sign on that you want is not something I recommend and is not possible without a much higher budget. Also, with phpBB, the admin section relies on authentication from within the site; having it accessible outside of the site would actually decrease security and make your forum more vulnerable. In my ten years administrating phpBB forums, I've never had a security failure (just keep a strong password).
Let me know if you have any questions, and I look forward to hearing from you!
Nathan