Hi, I have a Linux PC in my living room and an external hard drive with lots of music attached. I'd like to both hear it and remote control it from the other room.
Just wiring speakers from the other room to the pc in the living room is out of the question - it would require much tailoring craftsmanship which I don't have. A wireless solution is needed for speakers and remote control. The remote control should be something on a screen, not just a stick with buttons.
Do you have something like that set up? If yes, how did you do it? If not, how would you do it? What kind of hardware would you use? Which server/client software?
(I have found this thread and this thread about similar topic with no answers)

mpd, running on the server with an Android or iOS client works for me.

What android client do you use? I have only found mpdroid that does both control and playback but when playing the http steam there is a really long delay (around 30secs).

My Android tablets died but all I used was a basic control app and played back with something like VLC or a web browser. I didn't use the same device for control and playback. Let mpd run on one device with a stream open, listen to the stream on any other devices and have the controller connect to the server.

So what you did is basically run the client on the mobile device that you carry around through the house. But that means bad sound from a tiny mobile speaker, right (or having speakers in each room. Or having to carry around the speakers too)? Or having a mobile device plus external speakers I each room.

mpd runs on a device, that can be a server machine or a desktop with or without its own speakers.
mpd broadcasts a stream.
devices listen to that stream -- a laptop, a desktop, a Raspberry Pi,anything able to play a network stream over http.
Android, iOS or other device[s] controls this with the appropriate "app".

I am thinking more of a client device with solid speakers in each room (actually two rooms equipped is totally enough for me), and the only thing I have to carry around is the remote controlling device (which is separate fron the playback device like in your setup). But both devices should be able to play back the same stream simultaneously!

Can an Wlan Internet radio play an mpd stream or VLC stream? Can a VLC stream or mpd stream pretend to be an Internet radio station?

Icecast the audio from mpd and you can pick it up on anything.

You beat me to it -- I've used third-party icecast servers to create actual internet streams before for use in Second Life with mpd as the source. Though, in practice, I find that for some devices on a LAN the http stream from mpd will suffice.

This may sound stupid, but I don't know if an Internet radio will be able to pick up a stream from the local network, or if it's normally configured to go on the Internet immediately and only look for public streams? In that case, I'd have to make my stream public, wouldn't I? Is that what Icecast takes care of? Plus I'd need a public IP adress...
And that also raises the question if I want others to be able to listen to my music.

That depends...
If the "radio" only uses preinstalled streams or some closed-off library then, yes, you would be stuck with that.
However, if the "radio" will accept the URL of a station (I looked one up and found the menu page to add a stream) then mpd alone will suffice.
A Raspberry Pi or similar would work as a client also, or even a tablet or phone connected to a docking station or speakers of some type.
Basically, mpd will give you an audio stream over http that most devices, applications and "apps" can play and (from what I can gather) icecast adds things like song information to that.
By the way, I suggest mpd because it's designed to run on a headless server and be controlled by remote clients on a variety of OSs. It's also lightweight but does take some setting up to get right under certain circumstances like producing audio on the machine it's installed on when PulseAudio is involved.

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Merry Christmas everyone!
So I got myself this device for Christmas:
It is definitely able to play back a UPnP stream from the local network, but I don't know if it "has to" go out to the internet to look for http streams.
In any case, I want to use VLC as a source player and not Mpd, because I've had a very bad, super-frustrating experience with it - I spent hours and hours trying to set it up on my RPi and never got anywhere. So not interested in repeating that experience.
So my remaining question is how do I set up VLC to stream a playlist so that my webradio can pick up.
So far, no success. I selected the UDP (legacy) option in the GUI streaming wizard - the log (verbosity 1) keeps giving "connection refused" messages.
Do you think Icecast is a must here, or should VLC be able to do the job on its own?

VLC's never seemed to me to be designed for this kind of task as its collection management features seem tough to use. If you read something like the arch wiki on mpd once you get the hang of what's going on it isn't as difficult as it looks. The trick is to just read the bits about setting up music paths and inputs and outputs and ignore the install and server setup as that's done for you already on a Debian-type system*.
To broadcast from VLC you chose the HTTP option (I followed similar guidance to this and had some success) -- you're setting up an HTTP server listening for other devices, like your internet radio.

The point of both mpd, VLC, or some other setup with icecast is that they are set to output a stream of music in the configured format over HTTP so that any application which can grab data over HTTP, retreive the music and play the format being used it will work -- that should include whichever code is used on "internet radios". Or, to put it another way, provided you can enter a URL for a device to play that device has no way of knowing whether the source is a million miles away or on that same device**.

*If you want mpd to output to the soundcard on the machine it is installed on that can be a little trickier but it's only a few changes to make.

**In case anybody is thinking "surely it can tell by IP and subnet" I think that's a long an irrelevant sidetrack at this time.

I made it work with VLC over http! First success - very sweet. I found an option on the radio to add new radio stations with a prompt to enter an Url. I entered the local IP of my VLC-serving box with the port number and voilá!
It'll work if I have *only* the internet radio (why are you putting it under inverted commas? What would you call these devices?) playing now, but not for multi-room, because I get a considerable lag between source and receiver. VLC also seems to be aware of that: log says things like
"core warning: late buffer for mux input (12832)
pulse warning: starting late (-53792 us)"
and
"core warning: late buffer for mux input (10000)
core warning: playback too early (-50704): down-sampling"

My system is not Debian-like, unfortunately (I'm starting to have some doubts about this distro, as the PKGBUILD for vlc-git is broken, it doesn't even have icecast in its repos, and I basically have to build every other package from source), it's Arch-based.
If I can't make this lag disappear with VLC, I still might give Mpd another try (even though I know in my gut it's just waiting for the next chance to make me miserable...).

EDIT: it is not working for all the songs streamed by VLC. Some songs it just won't play. Sometimes the radio has trouble connecting to the stream.