The main reason why you are seeing two of the same images here is because you purchased what is known as an HDMI Splitter. Splitters work by taking a single input signal and, well, 'splitting' them out to multiple inputs.
Since from your original question you want to be able to have two desktops, I would suggest hooking up your laptop to a monitor that ...
As far as I know, it is a hardware switch inside the Headphone jack that will not allow for switching. Once the switch is engaged, the speakers are disabled. Have you thought about maybe using Bluetooth headphones?
You said of the older monitor "It's got both VGA and DVI connections" . You can get a DVI to HDMI adapter to do the conversion for you.
Here is an example:
You mentioned modifying what you have but that did not work and getting the actual adapter will very likely ...
What does a rectangle with two lines on the sides symbol mean?
A rectangle with two lines on the left and right side is a symbol used
to represent the VGA connector on many computers and devices.
Source What are all the symbols used by computers?
Most likely not possible.
My guess is your monitor detects plugged in jack and electrically switches output. There's no software at work, so it's not possible to change in software.
Check monitor's menu - if there's a related switch in there (highly unlikely), then maybe it's also togglable using DDC/CI.
As all suggestions have failed, I return to the only method not yet tried:
Use another good-quality cable (of another make).
If this doesn't work either, and if the TV is new, I would ask for an exchange
or (even better) ask for a refund. This TV looks to be defective.
If this is not possible, and if the TV is still under guarantee, then use it
to ask for ...