Does Seasmoke have bearing on Daenerys' dragon creation?
The dragon Addam of Hull rides, Seasmoke, resembles Drogon, Daenerys' dragon somewhat closely. There might be a family link even if it could be a coincidence or just the result of employing current VFX models for CGI. Rhaenyra's dragon, Seasmoke and Syrax, have been on Dragonstone and might have mated. Though we don't know for sure, the visual similarities between Seasmoke and Daenerys' dragons point to Seasmoke perhaps being Drogon's father.
Dreamfyre is Daenerys' dragon mother.
Additionally resembling Daenerys' dragons is Dreamfyre, Helaena Targaryen's dragon. Dreamfyre is said to be the mother of Daenerys' dragon eggs according to a theory. Dreamfyre is known to have laid several clutches of eggs, hence this argument is predicated on the knowledge that three eggs were taken by Elissa Farman, who subsequently fled to Pentos. Before heading to Asshai, Elissa is thought to have sold the eggs to the Sealord of Braavos. These pilfers most likely found their way into Daenerys' hands at last.
Fertilized are Daenerys' dragon eggs?
Daenerys' dragon eggs might or not have been fertilized. Like Dreamfyre, a female dragon may have laid them without mating with a male. In such circumstance, the eggs would be unfertilized and not hatch normally. But if given some sort of magical influence—like the one Daenerys went through upon Khal Drogo's death—there's a risk they might hatch.
What became of the dragon eggs Elissa Farman stole?
From Dreamfyre, Elissa Farman pilfers three dragon eggs. She left for Pentos, sold them to the Sealord Of Braavos, who subsequently sent them to Illyrio Mopatis. Later, Illyrio Mopatis gave Daenerys a wedding present—this one. How the eggs arrived to the Shadow Lands past Asshai is unknown. But the trip from Braavos to Asshai is a long one, hence it's likely that eggs ended up there following several exchanges.
How arrived the dragon eggs in the Shadow Lands?
Daenerys' dragon eggs arrived in the Shadow Lands beyond Asshai, but exactly how? When the Sealord of Braavos decided to sell them to Illyrio Mopatis, he might have been on a trip to Asshai. Purchased from Elissa, Future House Of The Dragon episodes or other media including books or games could reveal more about their path.
Why does Daenerys' dragon eggs being at House of the Dragon matter?
Daenerys' dragon eggs in House of the Dragon begs issues concerning her dragons' source. Dreamfyre could have laid them, or Syrax, Rhaenyra's dragon. It implies that Daenerys' dragon lineage originated in great part with the Targaryen family. The narrative gains an intriguing element from the likelihood that Seasmoke is Drogon's father. The dragon eggs also help to highlight the link between the past and the present Targaryen line.