Speaking from the cheap seats, Taiwan is obviously a sovereign country. They'd like to be in the UN. The UN apparently excludes them because of the influence of mainland China. So yes, they are excluded.
What makes them sovereign countries with a right to be in the UN is the amount of recognition they get from their peers. I don't have anything against the Taiwanese, but they are embroiled in a conflict just like the previous regions I described. Until they solve those issues one way or another and gain more international recognition they will not be able to participate in the UN, the Olympics or other places as a country.
One can imagine guerilla-controlled areas Columbia to be just like Taiwan, but I won't be travelling there...