I was finally able to fix all of the above problems and run linux succesfully on this device.
The 'solution' is a bit tricky though:
I simply formatted my linux partition and installed it all over again, this time with Unity (instead of GNOME).
After installing I found that reboot/shutdown already worked and the touchpad seemed to be better too. I then installed the 4.4.2 kernel succesfully and all of a sudden wifi was working and so was suspending. From Unity I then installed GNOME and now I have a working Ubuntu install with GNOME.
I'm back in Ubuntu! The issue was that the path to grub set in the BIOS after the firmware upgrade was wrong. Fixing the path fixed the issue.
To do this, press the F2 key while rebooting the Dell laptop, to bring up the BIOS interface.
Optional: In the menu on the left, first select System Configuration > SATA Operation, and select the AHCI radio button. (This step is required on my machine, because there are no RAID drivers installed for Ubuntu yet. If you have RAID drivers for Ubuntu, then you can choose RAID On instead.) Confirm the change of SATA Operation.
With the correct SATA setting already chosen, select Boot Sequence, and then click on the Add Boot Option button in the middle of the right pane. Name the new record (in my case — Ubuntu AHCI) and click on the [...] button to the right of the File Name field, choose a grub file for start-up. Select EFI > ubuntu > grub64.efi. Click OK.
Using the arrows by the list at the top right of the Boot Sequence pane, place your new Ubuntu Boot Option at the top of the list.
I had a new Boot Option with an unhelpful name (UEFI: THNSN5256GPU7 NVMe TOSHIBA 256 GB, Par) which had the same choice of file as Windows Boot Manager. I deleted this, and checked afterwards that I could still boot into Windows. Here is what my Boot Options look like now:
[✓] Ubuntu AHCI
[✓] Windows Boot Manager
Click Apply, confirm your changes, and then click Exit.
The machine should now boot into Ubuntu, just as it did before the firmware upgrade.
Best Answer
The Dell XPS 7590 ships with the Killer AX1650 WiFi module which, at the time of this posting, is not supported at install for Ubuntu 16.04+. After banging my head against the wall and contemplating reinstalling Windows to check if there was a hardware issue with the WiFi module, I finally stumbled upon the answer here.
The XPS 7590 is a beautiful machine, so I hope this helps someone.