Is there a way to get the USB-C from the Dell desktop to work?
The short answer, no. It would be rare for any desktop to provide more than 15 watts to it's USB-C ports. Why this is comes with the long answer that follows.
USB-C ports are required by the USB standards body to supply 7.5 watts minimum. Most computers supply more, either 12 or 15 watts, so don't expect more unless the computer lists USB-PD support from it's USB-C ports.
Supplying more power from USB-C out of a desktop computer is likely quite trivial. Inside most any desktop computer today is a power supply capable of providing many hundreds of watts but there's no real demand for more than a few watts of power from USB-C ports on a desktop. Any large peripheral, such as a printer or display, will simply plug in to the wall for power. Any small peripheral, such as a mouse or keyboard, will happily operate on a few watts of power.
Inside a common desktop computer will be a number of devices built to run on a standard 12 volts, such as cooling fans and hard drive motors. It would be nearly trivial for a computer maker to run 12 volts to a USB-C port but, as it turns out, this would violate the USB-C spec. USB-C allows for 4 voltages to be output to the USB-C port, 5V, 9V, 15V, and 20V. I don't know why the USB people disallowed 12 volts output to a USB-C port, I can only speculate.
If 12 volts were allowed on a USB-C port then there is a 3 amp limit, that's 36 watts. USB-C only allows more than 3 amps when supplying 20 volts, again I can only speculate why but I consider this a relatively wise choice. Even if 5 amps were allowed at 12 volts then that's 60 watts. If you look at the power supplies for laptops of the last decade or three you will find that the power supplies are often rated for output of between 15 and 20 volts, and about 3 or 4 amps. This is likely why the people that designed USB-C chose 15 and 20 volts as standards, and considered 3 amps minimum for a passive USB-C cable and 5 amps for active cables. There's simply a ready market for this and all they needed to do was package this into their already popular data and power standard called USB.
Putting more than 12 volts on a USB-C port means needing a power supply in the computer built to provide the 15 or 20 volts that USB-C allows, and currently there's little demand for this voltage but for providing more power to USB-C, and there's little demand for desktop computers to provide more than 12 watts from a USB-C port.
Just to show how popular 20 volt power supplies are in computing look at any Lenovo laptop, desktop display, and many numbers of peripherals, that they produced in the last decade or three. They've been using a standard 20 volt supply for many of their products, first with a yellow barrel connector, then with yellow "SlimPort" connector that shares a lot of features with USB-A and USB-C. Other manufacturers did much the same though perhaps with not the vigor to consolidate their hardware to a single power supply standard that Lenovo or Apple had. The Apple MagSafe connector was popular and provided 20 volts like their USB-C chargers today. Apple abandoned MagSafe for USB-C fairly quickly, other computer makers will allow charging with their old power supplies in addition to the option of using USB-C.
Would USB-A from the Dell desktop with a USB-A to USB-C cable work any
No. This is likely going to get you the same 2.4 amps that USB-C provides. Some USB-C ports will provide 3 amps, as will some USB-A ports. To indicate the higher power they may have a yellow or red color to the plastic in the port. When comparing 7.5 watts, to 12 watts, to 15 watts input to a laptop that wants to see 30, 45, or 60 watts, there's likely not much to lose or gain when picking a port. They will all be slow charging.
Do all computer monitors allow pass through charging?
No. This is a rare feature, and often comes with a high price tag since such displays act as a laptop dock as much as a display. Along with a display and power will come things like a web cam, speakers, Ethernet port, USB-A ports, and perhaps other useful items like a display output for dual display action, or a flash drive slot.
Perhaps just as common as a display passing power to the computer is a computer passing power to the display. I've seen Lenovo computers fit in a slot in the back of a display with the display taking power from the computer, and the computer taking power from a "SlimPort" 20 volt power supply.
What would be the best way to go about trying this (HDMI? VGA? DVI? Or
There is no good answer. For most computers there's no output port that is going to supply more than 7.5 to 15 watts from a USB port. Even so called "Rapid Charging" USB expansion PCIe cards I've seen supply only 15 watts to a USB port.