On the blocks that state to have the ability to set a custom image via GUI (the computers, TV, custom billboard and custom poster, this doesn't actually seem possible, right clicking does not bring up a GUI, but turns the screens on the computers on/off. It does nothing on the TV.
Right clicking with a wrench does nothing either.

This seems to be a core issue with MTS itself to my understanding.

As a test measure, try the following:

1. Place down your decor in the world as intended
2. Log out of your world (If playing on a server, it may be possible that you have to wait for it to restart.)
3. Reload into the world/server and try to set the imageURL

Alright, that appears to work, what image hosting sites would you recommend for hosting the images for these URLs?

Imgur was formerly my go-to, however changes in their external sharing policies have resulted in my decision to switch from their services to Dropbox.

Using the share feature you can change the initial URL from www.dropbox.com to dl.dropboxusercontent.com for a direct link, then paste that into ImageURL.