It does sound like the problem box is the mini box (which would be half the height of the other (main) box.
VM do tend to send engineers out to check problem boxes as sometimes they can be fixed without being replaced, however if getting the third box installed then they would need to recalibrate the signal levels and that would involve checking the current boxes and replacing them if necessary.
You won't be able to watch recordings if the main (fully working) box is powered off either at the wall or by the on/off switch on the back of the box. It needs to be turned on or in standby.
On both boxes go to Settings > System > Standby Power Consumption
There you will see 3 choices.
Fast start - the 360 starts quickly but is less energy efficient
Active start - more energy efficient than fast start but the box takes a little longer to start up
Eco start - the most energy efficient and probably the one you're using. It uses minimal power but takes a while to start.
The first two options are best if you have a mini box and want to watch a recording on it so I suggest that you make sure they are both on either Fast Start or Active Start
Regarding the problem box not being able to stream, this might be due to a lack of internet connection :- Does the program guide (EPG) on that box populate (update with current TV listings) or does it have a lot of channels showing, 'To Be Announced'? If most are showing To Be Announced then it's likely not connecting to the internet which would be why you can't stream on that device.
Both boxes should be connected to the internet. If you can stream on the main (bedroom) box then that suggests it has a working internet connection - if your VM hub is in the same room as the problem box it would be best if you can connect them together using an ethernet connection (a wifi connection should be sufficient there but a wired (ethernet) connection is always preferable. Whilst the recordings are held on the main box, VM's Horizon system (the operating system for both boxes) stores the actual links for the recorded programs in the cloud, so again the boxes need the internet connection to grab the links for the recorded programs in order to play them.
It sounds like the problem box hasn't been provisioned with all the TV channels if some aren't available. Which actual TV channels aren't available on the problem box? It's more than likely subscription channels such as Sky Movies/Sky Sports/TNT Sports. That should be an easy fix as VM can sent the correct codes to the box to enable you to watch those missing channels on it.