Can we just go ahead and recognize Taiwan as an independent, sovereign nation? This childish game of China's has gone on long enough.
Otherwise, it looks like pretty clear picture. The population identifies as (Taiwanese && not Chinese) at ~70-95%. The Sunflower movement, the election of the DPP and formation of the New Power Party, the Tsai administration's unprecedented defiance towards China.. all point the same direction, not to mention recent calls for independence-oriented referendums.
https://www.forbes.com/sites/jnylander/2015/02/14/strong-sup... https://www.npr.org/2018/06/20/616083178/born-independent-ta... https://international.thenewslens.com/article/113537
Yes, probably. The current government is pro-independence, and I think the majority of the people are as well.
As this point, the PRC is the party that's most strongly against an independent Taiwan.
> As far as I'm aware their claim is that they are the legitimate Chinese government.
Officially, yes, but it's very complicated. In the mid-20th century they were firmly committed to that idea, but now they mainly pay lip service to it in order to walk a very fine line with regards to the PRC.