It looks about right and you can use any input that has composite video for the karoke machines. The labels are just for convenience and don't correspond to functionality.
You don't switch video on a receiver. There is a well defined order and the manual should describe it (all receivers works pretty much the same way). Because you have multiple video formats and the receiver does not transcode from one format to another you will need to make multiple connections to the TV and switch the video source at the TV.
HDMI in goes to HDMI out, component video in goes to component video out, composite/s-video in goes to composite/s-video out ('monitor out'). So with HDMI for the DVD player and an HDMI cable to the TV, the TV has to be on the HDMI input. When you switch to the input for one of the karaoke machines, you'd have a composite cable from monitor out on the receiver to a composite input on the TV and will have to switch the TV input to the one you connected to the composite video cable to.
A universal remote with macro capablity can make all that switching into a one button affair.