Landing in an airport other than destination due to bad weather

AFAIK as it is an international flight, the Montreal Convention should apply. It clearly states that the passengers have a contract to be transported from one point to another one

Extracts:

Article 19 — Delay The carrier is liable for damage occasioned by delay in the carriage by air of passengers, baggage or cargo. Nevertheless, the carrier shall not be liable for damage occasioned by delay if it proves that it and its servants and agents took all measures that could reasonably be required to avoid the damage or that it was impossible for it or them to take such measures.

IMHO, it means that it depends on what was really said (and possibly written) at Tbilissi. If they said that you were on your own, full stop, they did not respect the contract and you can use MC to claim the full compensation of the damage in the limit of the carrier liability. But if they proposed to continue the travel via Warsaw (yes even if it was the start point) but simply could not say at that moment when you could reach Yerevan, they reasonably can argue that it was the best that they could do at that time.

If they did propose you to continue the travel by your own means and that they said that you would receive a compensation, fine, you can use that to claim for the compensation, but it looks like it is not what has happened.

If you choosed to stay in Tbilissi, you willingly stopped the travel. You could argue that you had to do that, because the carrier was not able to transport you to the normal destination in a delay that was acceptable for you. Maybe you could ask for the compensation for the delay that you could have asked if you had choosen to go back to Warsaw and had reached Yerevan with another flight from same company. But I would consult a lawyer before starting any legal action here...

You can evidently claim for a refund on a commercial agreement bases, specially if you can explain that you often use their services and would be sorry to now use another company.

I'll go on a whim and try to answer as best as I understand the question. If I misunderstood something, please let me know and I'll either change the answer or delete it if I can't answer the question any longer.

The airline did offer to fly you back to a location where they could provide you with any necessary support. The reason they couldn't give you an answer about when they'd be able to transport you from Warsaw to Yerevan was because they can't predict the weather. If you flew with them back to Warsaw, they would do their best to get you to Yerevan as soon as practical. Although for weather related delays it's not strictly required, I suspect they would even provide you with some meal vouchers in Warsaw airport. The airline did not refuse to continue your journey - they just couldn't do it at that time due to external circumstances - and they were trying to do their best.

It is very expensive for an airline to land a plane in a non-destination airport, for any reason, plus keeping the plane at the airport also costs them money, therefore they needed to get the plane back in the air as soon as possible. Very likely the only reason they even landed was that they didn't have enough fuel to fly back to Warsaw.

You effectively voided your right for any refund or support by voluntarily aborting your travel and leaving the plane before reaching your destination.

While you can still attempt to write to the airline and request some compensation, technically they are not obligated to provide any - and I would be really surprised if they did.