“if Republic of China want to become Republic of Taiwan, they probably should publish the declaration of independence”

They don’t have that choice. While independence is quite popular in Taiwan, the PRC has made it very clear that they see any movement toward Taiwanese independence as cause for war. Going so far as to fire literal warning shots over the island in 2022 and 1996.

Depending on where the burn is you can just put the burnt part in a container with cool water instead. It’s much more practical to walk around with your hand in a cup than to be standing next to the faucet.

But the PRC is not a direct continuation of the Qing.

The USA can’t lay claim to Great Britain just because they used to part of the same country before the revolution.