I'm sort of the same. I want my phone and computer to just work, leaves me with more time to play with more important stuff like servers, virtualisation, networking and so on.
With regards to the NAS, have a look at the HP Micro Server. They cost about £250 but there is a seemingly permanent £100 cash back offer, so it works out about £150. Whilst the cpu is low spec and it's only got 2Gb ram / 250Gb hard drive, you can up the ram to 8Gb for £30 and it's got four drive bays (no optical drive though).
Get one then install the free VMware Hypervisor and install OpenFiler (available as a VMware appliance). You've now got your NAS and it will be able to run some extra OS's on top. I've got a copy of XP running on it solely for downloading, plus I've normally got a few web servers running on it for development purposes. Also got SBS 2011 running on it. Handles it all fine.
Surely all those VMs just take up resources though? And I thought you wanted an out-of-the-box solution, not something to tinker with
Anway, you could just use something like FreeNAS or Nas4Free as your OS which will do all of that (except SBS 2011 of course). Pre-installed services include usual NAS stuff like SAMBA, Rsync, FTP, SSH etc. As for your extras, it can be used as a webserver, uPNP media server and there's also a torrent downloader in the latest version.
With it being nix based, it's very easy to add additional packages like jdownloader.
And Barry, both FreeNAS and N4F support hardware & software RAID and ZFS which will be all you need for data integrity purposes. It supports Time Machine too, but I've never used it.