Is it somehow possible to update Windows Server 2016 from 1607 to 1703 or higher without performing an upgrade? Or, if an upgrade is necessary, can it be an in-place upgrade?
Are you upgrading with same Server version ie datacenter, standard, essentials? If not it will require fresh install.
What you are saying contradicts the information on this Microsoft page: FAQ on Windows Server, version 1709 and Semi-Annual Channel Are you saying you have some sort of alternate method for upgrading that sidesteps this requirement?
Server 2016 is Server LTSB (with GUI), Server 1703 is semi-annual Core-only. Both are different products.
Is there some way to slipstream the 1703+ updates or otherwise modify the installers to integrate the updates into Windows Server 2016? A clean install is not a viable option for me.
Server 2016 is RS1 branch (1607). Only updates for that branch will work. Updates from other branches are incompatible and cannot be ported. Upgrading to Server 2019 LTSC would get you onto the RS5 (1809) branch.
Is there a way to perform the upgrade without restarting? I am using a remote machine rented from Paperspace, and if I perform a restart during the upgrade process the machine will become inaccessible.
No. Upgrades to any Windows OS require multiple restarts. Plus, Server 2019 LTSC requires a new license and will not work with the 2016 LTSB one. If you are using RDP, the machine should become accessible after the reboot(s), again.
I can't seem to do an in-place upgrade to Windows Server 2019. When I use this: C:\Windows\system32>dism /online /get-targeteditions I get this: (The current edition cannot be upgraded to any target editions.) Is there any way to change or fix that? DISM says my current edition is ServerDatacenter.
Server 2016 to Server 2019 is not an Edition (SKU) upgrade, like Server Standard to Server Datacenter, for example. You need to do a full OS upgrade, using a Server LTSC 2019 ISO.