I have news, actually I have POSITIVE NEWS!  This is a bit awkward, but it looks like I could solve some of the issues in a very simple manner:  I realized that mouse and keyboard dying after a minute resembled my ethernet inside the TB16 also dying after some time and that all of that points towards power management. So I registered all three, the mouse receiver, the keyboard receiver and the ethernet adapter inside the dock on the blacklist of the power management to prevent them from ever getting auto suspended.  Et voilá: I have been using my mouse and keyboard for a few hours now without any issue, also the ethernet suddenly appears to work fine so far, though I only registered that inside the blacklist some 30 minutes ago. So I will have to monitor that a bit longer.  What is strange about this:  Mouse and keyboard did work reliable before updating the BIOS. No changes to power management done by me before. The blacklist already contained USB HID devices before, which apparently was sufficient before the update. Now, after the update, it appears as if mouse and keyboard are not registered as HID devices any more which explains why the old blacklist does not cover them any more... Dell might want to take this as a hint to either fix the actual cause of the issue or to contribute to the Linux power management to prevent USB devices plugged into the TB16 to be auto suspended.  The audio jack on the TB16 backside still does not work and most likely HDMI still is not usable. But lucky me has an old panel with a display port.  :-)
After more or less having given up on Dell being able to provide a working setup with their own products I today found the new BIOS firmware version 2.1.0 and gave it a try.  My setup: xps13 (9360 running Ubuntu with kernel 4.10) with a TB16 with updated firmware.  I upgraded previously and spend quite some time trying to solve various issues. Had different docking stations from Dell before and ran into huge issues. That got better with the second TB16 I received (the first was some early bird model apparently and did not work reliable at all). With the current TB16 which I have been using for month now I can boot with a connected dock. Hotplugging will sometimes work, sometimes not. I can attach a screen through the dock (DisplayPort) which works ok-ish, did not even bother to check HDMI because of endless reports that it does not work. Back side audio does not work but the front one, ok, not nice but who cares. The ethernet is unusable, works for maybe a minute, then starts to drop and pick up the connection all the time. Unusable, so I have to use WiFi all the time. Great for a docking station... At least USB in the dock worked for my keyboard and mouse, so for slow devices.  After upgrading to the current BIOS (2.1.0, had 1.3.2 previously) things have not improved, ethernet still unusable. But to my horror I found out that USB on the dock does not work any more reliably. Keyboard and mouse are recognized and work for maybe a minute, then they die. Replugging them reactivates them, again for maybe a minute. So unusable too now. Great!  I have to plug both USB receivers into the Laptop itself. Exactly one of the reasons why I invested lots of money into the dock: to not having to attach all sorts of plugs manually.   This is sooooo frustrating. Apparently Dell gave up to find a solution to the problems with their own hardware. Why is there a "project" and messages that they are working on a solution, when nothing happens even after month? Frankly I doubt anyone actually works on those huge issues. To me this looks like Dell simply gave up on fixing their own hardware they sold for much money. Very sad.  Or does anyone have a hint for me that helps solving at least the annoying new issue with the USB ports on the TB16? Can I downgrade to a previous version of the BIOS (1.3.2) or will that ruin the system? 
Updating BIOS to 2.1.0 makes support for TB16 even worse