Faramir Jones’ discussion and explanation of Thorin’s legal position and Bard’s legal condition appears to me to be very complete.
To add to it I will only note that Thorin may legitimately see this as an attempt by two large armies, one of desperate Men who have lost their town and ¼ of their number to the dragon, to bully him who has only 13 companions.
Thorin’s question about what share of the treasure would have been paid to Thorin’s heirs if the Men had found him dead is evaded by Bard. Thorin is implying that the Men (and Elves) would have paid nothing but have simply confiscated the treasure. I suspect Thorin is quite correct.
From Rateliff’s
The History of the Hobbit, page 656:
After Fram of the Éothéod [Horse-folk] one of the ancestors of Eorl the Young, kills ‘the great dragon of Ered Mithrin’ [Grey Mountains], the dwarves claim his hoard, which had been stolen from them by the dragon (as evidence of which, note that Merry’s horn, which came from that hoard, ‘was made by the Dwarves’; LotR.1014). Fram ‘would not yield them a penny, and sent to them instead the teeth of Scatha made into a necklace, saying, “Jewels such as these you will not match in your treasuries, for they are hard to come by.” Some say that Dwarves slew Fram for this insult. There was no great love between Éothéod and the Dwarves.’ – LotR.1102.
Thorin certainly could have acted differently according to medieval custom. A counter-offer somewhat less than Bard’s ultimatum would have indicated willingness to parley without appearing overly weak. But certainly Bard might have, at first, been less obviously cocky. Perhaps Thorin’s insistence that he would not negotiate with an overwhelming army was an attempt to open negotiations with a smaller number, although I doubt it. Bard still might have tried to take Thorin up on it, but he didn’t.
Bilbo notes, “these were fair words and true, if proudly and grimly spoken; and Bilbo thought that Thorin would at once admit what justice was in them.” It seems that even Bilbo with the phrase “what justice was in them” thinks that Bard is going a little too far, although Bilbo is more shocked by Thorin’s refusal to negotiate at all.
The Great Rings I do not see as an issue. Thrain’s Ring had been long ago stolen by the Necromancer (Sauron) and is not in Thorin’s possession. Also when he first wrote this Tolkien had not yet invented the Dwarf rings.