There's one thing I forgot to consider in my original reply and I'm sorry for that. With TrueNAS you'd probably have to copy your data off of the existing drives to somewhere else because they will have to be reformatted to create a ZFS pool. I don't know if that is practical for you, so please don't feel like the following is something you must do or anything.
I think you're doing great. Sorry for the late reply. To answer your questions:
TrueNAS Scale is an OS that is built on top of Debian. Using TrueNAS makes set up simpler to set up, but you could implement what you want with a Debian install, but if you were to install TrueNAS, it would replace whatever existing OS you have installd.
Yes, TrueNAS would manage your filesystem. It can manage your hard drives for you. Its UI isn't too hard to understand, and it can be accessed and managed through a web browser on your Laptop.
TrueNAS has some software packages in the form of docker containers, they are managed through the TrueNAS UI. You can browse them though their website here. My advice with these apps would be to set up your NAS with all of the drives in storage pools first before installing these. If theres something you want that isn't supported, TrueNAS can also set up Virtual Machines, and you can use one of those to run those services, provided your CPU supports it and its turned on in the BIOS. If you go need to go down this route, you will have to set up a bridge network in TrueNAS in order to get the VM to communicate with TrueNAS over your network, but that's not particularly hard or anything.
You will need to run TrueNAS on its own computer, yes. What I was suggesting was installing TrueNAS as the OS on your Desktop. Idk if thats practical for you or not, since doing so would need you to wipe everything on the boot drive of your Desktop, so idk if you have a place to copy any important data off of it to.
In terms of comprehension, yeah I think you've got it.
I think a NAS system would handle your caching idea for you, if I'm understanding you correctly. Having a good file sharing setup over LAN, whether its using NFS, or Samba would allow you to mount a folder from your Desktop on your Laptop and access them.
For files that you want to have access to when not on your home network, you could set up a folder that Syncthing tracks on your Desktop, sync that with your Laptop, and then have access to them that way.