Do I need at least 2 RAM slots full for proper functionality?

No. You can upgrade by the each DIMM. And if yours EVER got to the login screen, two are not needed.

However, your one must be exactly the correct ONE, and it can not incur any errors during the Power-On Self Test, or your one slot will be declared empty.

The Mac Pro with Xeon processor uses Error Correction Code RAM. The error correction logic is used very aggressively at startup. ANY errors, correctible or not, will cause the slot producing the error to be declared Empty, and MacOS will not use it. However, after a shutdown/restart, all DIMMs are tested again, with no history, so it may pass the next time. But that does not make it good, it just means you fold it, and it may shut down with a distinctive kernel panic, machine check if an uncorrectable error occurs.

There is very little testing that has been done with just one DIMM. It may be preferable to have two or more for no other reason. Remember that you cannot mix 16GB and larger (Registered) with 8GB or smaller (un-Registered), they do not play nice together.

OWC policies allow you free exchange, like for like, of any RAM that displeases you.

But the DIMM you have does not seem to be working correctly, or perhaps you have an additional issue.

If you could examine the kernel panics saved at:

/Library/Logs/DiagnosticReports

you could look for a file with date&time and ending in .panic. If it is a kernel panic, machine check, detected by multiple processors, it is very likely a RAM error.

User Tip: Mac Pro and Error Correcting Code RAM Memory

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If you place same-size DIMMs in slots 1&2 or 3&4, they will interleave and run slightly faster.

All four runs very slightly faster still, but not likely enough of a change to be perceptible in real-world uses.