Can I get a NAS that is read as a network storage device and functions just as internal hard drives to clients?
That is hard to say. Depends how you define 'just as an internal drive'. Normally a NAS functions as a Samba server, which means the data shows up in a share. If that is 'like an internal drive', then yes.
If you want a level deeper, then you should need a NAS which supports iSCSI. It means that a partititon (or a virtual partition) is seen on the client as a SCSI disk. The client should support iSCSI also, of course. A problem is that only one client can use the partition at a time (unless you use a special filesystem which supports more than one host simultaneously. The client
has to support that, as the NAS only offers a 'disk'.
An iSCSI device will in most cases have a higher latency than a directly connected disk.
Would I just need to connect it with USB to my desktop but also have it connected to my router for other devices?
In case of Samba shares it's not possible to have a disk mounted via USB and simultaneously shared via Samba. (Unless the USB host shares it).
I understand they generally come with integrated operating systems
come with an integrated OS. It's not just a disk which coincidentally has a network connector instead of a sata/usb/scsi/... connector, it's a server.
I would be able to assign it a static IP and name through my router, right?
If your router supports it.
Can I give it considerable security, even in a trusted home network? Forced encrypted communications etc.
AFAIK Samba nor iSCSI support encryption on the data streams, so no, not out of the box.
Would it be feasible/possible to, say, have a virtual machine/console emulator on the NAS, do all computation on the client, and write only to the NAS?
I don't know what you mean.
Have encrypted containers of sizes larger than 100 GB and decrypt them on the NAS, rather than sending the file to the client?
If you mount an encrypted container on a client, the file is not send to the client, but the client just reads that parts it needs.