issue with HDMI port & audio jack

If you have an add-in video card, make sure the monitor is connected to the card, not to the onboard DP port.  An add-in video card would be in the area marked "2" on the back panel, shown here. 

Did all the drivers get installed, including Intel Chipset and Intel Management Engine, RealTek Audio driver, video drivers, Intel and either AMD or NVidia, if you have add-in video card? You can enter your Service Tag on the support site to find the drivers specific for you PC's hardware. (Don't post Service Tag here.)  NOTE: Intel Chipset driver is listed under "Utilities" on the support page, rather than under Drivers. So easily overlooked.

Did you install the monitor's own driver?  If it's not a Dell monitor, you'll have to go to its manufacturer's site to look for the right Win 10 driver.

When that's all done, go to the Win 10 Sound screen and configure Output correctly to use the soundbar. Probably should be set to "Realtek Audio".

This PC also uses Waves MaxxAudioPro as sound controller, so did that get installed?  After you install Waves, power PC off normally. Then disconnect soundbar with power off. Reboot PC and when you're at the desktop, plug the soundbar into rear (green) port. Now wait for Waves to pop up and ask what you connected. Select the best option from its drop-down menu.

When that's done, open Waves Control Panel and click the Playback box. When that opens, you can click the ? near the top to see what each control does.  And make sure Waves sees your soundbar correctly on that screen, in center, toward the bottom, or select a different option from it's list.

I don't have an add-in video card, only the onboard HDMI and VGA.  Is getting an add-in card the simplest solution?

Update: if I plug the monitor into the VGA instead of the HDMI, the sound bar comes alive and plays audio with no problem.  But I would rather have the higher quality graphics, so if the onboard HDMI just doesn't work for whatever reason--and the BIOS won't let me do multi-display--then it seems like getting an add-in card is the way to go?

Were all drivers, Waves MaxxAudio Pro, and Intel chipset etc installed?

Are you sure the DP/HDMI cable is the right orientation, eg HDMI(PC)>DP(monitor)?  These cables generally work in only one direction, and can't just be flipped around. Double-check the cable to see if its orientation is marked, either on the wire and/or on the connectors.  (Connectors may have an icon for PC or monitor on them.)

Can you test this monitor and same cable via HDMI(PC)>DP(monitor) with a different PC or a different monitor and cable with this PC?

VGA doesn't do anything with audio. So I don't see how using a VGA connection to the monitor would enable the soundbar.  

All drivers and software are up to date.  I read somewhere else that DP-HDMI cables are one-way.  I tried the (brand new) cable on another PC and got no monitor signal.  I must've bought the wrong kind....

I said above that video cables typically only work in one direction. So that would be a simple explanation for your issues.

Go get one that's the right orientation this time...