I urge you to break up your wish list into more managable and approachable pieces. My thinking sees at least two "servers".
Take an olde tower PC and load your favorite server distro. Get your in-house interconnections working solid so that you don't fight that as you add new server features. Make it work in-house for file sharing and printer sharing and backup and such. Add an in-house web server -- the XAMPP suite does most things well without your need to track down a lot of parts. Do something in-house with your web server content. I'd consider one of the suites like Joomla or WordPress.
Fetchmail will grab email from a list of accounts and bring it all to the local server. You'll need some local mail transfer suite like Postfix and then SpamAssassin with ClamAV for scumware defense.
For example, one can find several articles about setting up a home-network "media server", so pull your media service wishes out of the long list. There are even entire distro's devoted to that like Mythbuntu http://www.mythbuntu.org/
You mentioned that you wanted "remote access for family members in other areas". If you want your in-home server available on the public internet say http://www.myHomeServer.com
then there are all sorts of issues about security and networking involved. On the other hand if you simply want someone "out there" to be able to fetch a file or two from time to time, that is another whole barrel of snakes. Both have well understood solutions, but with some learning curve and tinkering on your part.