Question 1: Since you are able to get the dock to work sometimes, your 5480 must have Thunderbolt. Models with Intel U or H series processors and only integrated graphics do not. The System Configuration -> Thunderbolt Adapter Configuration page includes an option to Enable Thunderbolt Adapter Boot Support. It is disabled by default. The first thing I would try is to enable this setting.
Question 2: According to this PDF, your dock does support 2x4K displays at 60Hz if your system supports DP 1.2. Note the double **, which reveals that Question 3 is correct and you need one display connected to the Thunderbolt Type-C port on the dock. There are two Type-C ports on the back and only one is TB. That means that for Question 4, yes, you need either a USB-C to DisplayPort cable or a USB-C to DisplayPort adapter that you can connect to your existing cable. If you choose an adapter or cable from a reputable company, I'm confident it will work per this post about about the WD19TB. This should fix the resolution issues on both your machines.
Since you have all Dell equipment, you have the best dock for your needs. The Dell docks have custom power delivery that works only with their laptops, and any third party dock you get will still charge but with less power. Finally, you should definitely not buy a Dell USB-C dock because none will work with dual 4K displays and both your machines appear to have TB. The D6000 is a decent dock for productivity use, but in high performance scenarios, the WD19TB will be better since it is not DisplayLink.