What do we know about the current Orsinium? It's located between Hammerfell and Skyrim now, yes? But who rules it right now? How big is it?

Is there any small pieces of information that can tell me more about this?

*Waits for Malak the Orc

LOL. As of Skyrim it is now located in the mountain range between Skyrim and Hammerfell. Bretons and Redguards have sacked it countless times. And Orcs are interesting as hell. But as I wrote earlier, I'll defer to the experts on the subject.

From what I can gather it's a city rather than a country, located in the wrothgarian mountains in high rock. Ruled by orcs. It's like a big stronghold.

Note- I am far from a lore buff, so could easily be wrong, but I am an orc fiend.

Nothing much. We know from Orsinium and the Orcs that Gortwog didn't just take a piece of land, but claimed it through local and Imperial rights: I've a feeling that this kind of legitimacy is important for the Orcs, so they would probably rebuild in similar conditions: on a piece of land that is rightfully their own, under a king of proper lineage. I've always assumed that the old Orsinium was built over a mine, which would explain its forges and famous iron walls. Presumably the current one would be close to or above a mine as well.

The empire acknowledges the city and seems to have a policy of sending Orcish remains back there. Perhaps there's a communal catacomb beneath the city? Some kind of Orcish public cult of the dead?

I imagine it would be a "city-state".

The current one, Nova Orsinium, is located in the Dragontails rather than the Wrothgarians. They relocated after it was destroyed yet again.

I don't recall much more than that, but I don't know if there is much more than that. Information is sparse.

I would think so too. In my opinion, it's an independent city-state with its own government established, free from Imperial rule.

I imagine that regardless of whether it is important to the Orcs for other reasons, they know well that it will be important to the Empire. Simply migrating a la the Ra'gada is likely to provoke the response of several Imperial Legions, as well as whatever local forces could be mustered. Without an invasion, per se, the Empire has little to officially respond to, and as you rightly say, acknowledges the city. Any local resistance may not even materialize, without promises of Imperial support. Stability is rather important in the face of the Thalmor, after all.