I have no experience w/ nMedium, but have been using the JRemote app on an iPad for months. I love it. A recent purchase of a Surface 2 has me very interested in the ability to drive my server remotely, and I'd be very interested in an app that provided me that capability.
EDIT: Adding my "votes" on most important items.
I typically play music in one of the following ways.
Select music from one of my Library Views. I have my music separated from my wife's through use of a custom field for Owner. That allows me to browse music either as "Album by Owner" or "Artist by Owner". So being able to see those views through the remote app would be great.
The second major way I select music is through a small number of Smartlists that I've created to randomly select albums.
The third major way I select music is by searching on Artist, Album, etc.
Once I've done that, I usually just use play, pause, skip, etc. Seeing cover art would be great.
I don't really use Play Doctor, and the other items you list such as videos, creating/editing playlists, tagging, and re-ordering items in Playing Now would just be gravy for me.