RE: HDMI Output only detects 2 channels not 5.1
The configuration screens in your screen shots are properties for the speakers, not for HDMI audio. Your laptop has only has stereo speakers built-in, and only has a port for connecting external stereo (or 2.1) speakers, so naturally there is no setting for 5.1 speakers. A 5.1 speaker system is a set of 6 speakers, which does not have a DAC. A 5.1 speaker system, when connected to a computer, depends on the computer's DAC to send out 6 channels of analog audio. But you cannot connect a 5.1 speaker system to your laptop, so there is no configuration for it.
In the past, Dell has offered a number of laptop models that were compatible with a 5.1 speaker system. Those models had 3 audio ports -- 2 headphone jacks and one mic jack. When configured for surround sound, the mic jack became a 3rd output jack, for a total of 6 output channels, 5.1. The mic jack became the output port for the speaker system's 2 rear speakers. Here is a screen shot from a Dell XPS L502x model:
should I purchase a laptop that has an on-board sound card capable of outputing 5.1
If you have a 5.1 speaker system and want to connect it directly to a laptop's analog audio ports, then yes.
or 7.1 Audio?
I don't think Dell ever had a laptop model that supported 7.1 analog audio. For that, you would probably have to get a desktop system with a 7.1 sound card.
However, all of that is irrelevant, because your thread is about the HDMI audio, and when you make HDMI audio the default device, then the speakers are disabled so it doesn't matter how they are configured.
I upgraded my Realtek Audio drivers
Please read the HDMI Audio FAQ if you have not done so. HDMI is controlled by the graphics driver, not the audio driver.
According to Dell's documentation, the XPS 13 9350 has Intel graphics. You can find all of your graphics drivers on the XPS 9350 html page -- scroll all the way down to the Video category. The Sept 7 version of the driver says it fixed "HDMI", but no details about what was fixed. [Note: graphics is outside of my area, so I can't advise you on which driver is the correct one for your model. The driver list has both Intel HD Graphics drivers and Intel Iris Graphic drivers. I have no idea what is the difference between them, if any.]
Where I should be able to select either stereo, 5.1 or 7.1 Audio.
HDMI audio is in the form of a raw digital signal. The surround channels are encoded in the signal, and it is up to the TV or surround receiver to decode the digital signal, and to convert it into the analog channels. I don't know of any software in the laptop that would affect that process, after you make HDMI Audio the default device, but if there is a configuration setting, it might be in the graphic's control panel. My Latitude E5420 laptop has Intel graphics that is older than yours. I can't find any HDMI audio configuration in my graphics control panel.
The documentation for the XPS 9350 shows that it does not have an HDMI port, so how are you connecting the laptop to the TV or receiver? Usb-to-HDMI adapter?