RE: Latitude 7480 External Displays
Yes, it will support USB-C to HDMI or DisplayPort cables/dongles. You could even use a dongle plus a DisplayPort MST hub in order to drive multiple displays off of that single USB-C port if you wanted to, and you'd be able to run display configurations that were within the bandwidth of a single DP 1.2 output, which is enough for 4x 1080p displays or 2x 1600p displays. However, you also have to stay within your GPU's display max, which is 3 on that system. Note that the configurations I just described would NOT be possible if you were connecting through a USB-C DOCK, since in that case the dock would be using the USB-C port for both display and USB data, which cuts display bandwidth in half. This is why USB-C docks only handle 2x 1080p.
Additionally, if your 7480 has Thunderbolt (which is optional on that model), then after installing the latest BIOS and Thunderbolt firmware update, you could also use Thunderbolt 3 to Dual DisplayPort adapters. That solution is even better because Thunderbolt would give you the bandwidth of TWO full DP 1.2 outputs rather than one, so you could for example run 2x 4K displays.
And yes, using USB-C and HDMI simultaneously would also be fine.