Nothing really exists like that yet. As for Boxee it does come close but Boxee Box by the looks of it is at the end of it's support, Boxee team have gone working on Boxee Box 2.
Boxee Box 1 will give you
* iPlayer official nothing else
* Unofficially there are some other plugins out there for UK stuff but I don't keep track but don't expect much. Navi-X is also another avenue.
* Spotify client
* Good media player and automatic jukebox
* Iomega model comes with HDD built in, but you can share out USB drives on D-Link model too.
* Boxee has web browser you need a mouse to control it. Whether those sites you mention work under Boxee I couldn't say.
* No Android remotes.
Another option is set top box running Android which could give you;
* Some do come with a HDD bay but you will have to setup a
Samba server yourself.
* iPlayer and any other catch up apps available under Android.
* If usable through Chrome web browser with no plugins then should work through Chrome on Android.
* Spotify app is available for Android.
*
Droidmote does this but it's just another cursor like a mouse it's not a separate UI.
Be aware however that your taking something intended for another function and tinkering with it to do something else so things can get messy. Also the media players on Android set top boxes are extremely basic, most formats are covered but the players are not as polished and lack features compared to dedicated media players. The Boxee Box is still leagues ahead as a media player however Android gives you a lot more flexibility in what you can do with it.
There are also hints being dropped on XBMC forums about something to do with Android but who knows what that actually means and when it will show up.
As to an Android box well there is the
Mele A2000 which takes a 2.5" HDD, there's a dock flap on the top of the player. This device also has a GUI for browsing media via USB, SATA, SDcard, Network shares via remote like a stock media player, it's very basic but it works. The newest firmware runs Android 4 reasonably (it's no speed demon) and will probably get updated to Android 4.1. Work is also being done to port XBMC to these devices running via SDcard. Owners thread on avforums over
here Geniatech ATV3000, takes a 3.5" HDD internally, these are based of Amlogic chipsets, they lack any sort of special GUI for browsing media with a remote, your expected to use wand remote or mouse. Awkward browsing of network shares and these are based on the 8726-M1 Amlogic chips which Geniatech are not supporting very much any more as they were intended for Android 2.3 and Android 4 doesn't run that well on them. They haven't announced a replacement for those models yet with the newer 8726-M3/MX chips.
I've used both of those boxes and would lean towards the Mele A2000.