The below solution has now been implemented. It did work.
Can now plug in a laptop, access internet, play video, and have sound play via home theatre system.
This technical information is key to why this set up does not work.
HDMI-ARC uses two pins on the hdmi cable an unused one and the hot detect pin, it is not part of the normal data stream and does not use the normal pins.
It's basically a coax spdif connection in reverse.
The confusion has arisen it would appear, when the home theatre box manual mentions ARC return feature associated with the HDMI port in a way suggestive of it being essentially a two way hdmi port.
It would seem ARC does not deliver audio data by the standard way of hdmi video/audio data streaming protocols, or even by the same internal physical pins.
A TV set will return ARC signal, as it is designed to use this different standard for the return audio.
Apparently, ARC is part of the hdmi standard though, as was commented on another site.
ARC is part of the hdmi standard .... it was an addition to HDMI to allow a TV to return audio back to an AVR., to replace a spdif coax or optical cable.
This missing technical understanding re ARC explains why a suggested solution was to add to the hdmi cable from the laptop at the theater box end, an hdmi audio extractor box, with a audio fibre Toslink (for 5.1 surround sound) or RCA for stereo sound, input to those available audio input connections on the back of the home theater unit.
The situation has been confusing. It would seem reasonable to assume the connection functionality of the home theater box would be inclusive of plugging in a laptop device, one accessing the internet for various audio visual media. Hence, when not working, the natural assumption is that there was something wrong with the numerous setting options, which is where I began, or the functional capacity of the hdmi laptop port or laptop settings.
However, it seems not so. The danger of unconscious assumptions!
A second part of the problem is that it the hdmi splitter should be an hdmi switcher, so that the hdmi video data can be selected as coming from the theater box or the laptop.
Diagram of expected solution set up below.
Credit and thanks to the guys on the Whirlpool site, as well as those commenting here.
Best Answer
No, it does not, I fear: the DisplayPort is an output only.