Your last question first: most EU countries, certainly those in Schengen, have no internal border checks at all - it is the same as crossing from one US state into another. But for your list, only Slovenia and Hungary are in the Schengen zone. (And Greece I guess, but it has no land border with any Schengen country, only ferries and flights.)
In practice, I can't imagine any country other than those bordering Georgia actually caring about the export declaration, as long as you have proof of ownership of the vehicle.
How are you intending to get from Romania to Armenia? I expect the Black Sea ferries are not running because of the war, so you're pretty much stuck going back via Bulgaria and through all of Turkey into Georgia (the Turkey-Armenia land border seems to be closed), and into Armenia from there.
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