Do each of the permanent members of the UN Security Council have veto power over new membership applications? If so,
To answer the question of if a Permanent Member of the UN SC can use their power to veto an application for Member State, then the answer is yes, they can.
However, that said, since 1972 China has only used its Veto 9 times. And only 1 was regarding the admission of a new Member State. In this case, it was an application by Bangladesh, and occurred in 1972.*
To answer your other questions:
1.Doesn't China's veto power mean that it would most certainly veto any membership application by Taiwan to join the UN, and therefore it is impossible for Taiwan to join the UN?
Interestingly enough, it's not China (directly) blocking the applications. As it stands, the main roadblock to Taiwan becoming a permanent Member State isn't China's position on the Security Council, but rather the General Assembly itself.
The process of application to become a member state is described here.
A lot of Taiwan's applications have not made it to the UN SC, rather being rejected outright by the Secretary General under the context of the General Assembly resolution 2758, which recognises the PRC as "the only legitimate representative of China to the United Nations" and reinforces the UN's 1 China 'policy'.
This is a rather good article on it, and also answers your second sub-question:
2.Is there some viable alternative strategy for Taiwan to join the UN
Short answer is, at this stage, no. While they are trying to work with UN organisations, Taiwan is not likely to become a recognised Member State of the UN.
*Here is a full list of the UN SC vetos.
The 1992 Consensus is currently working for both countries politically speaking because economically the major trader partner of Taiwan is China; the Consensus is something debated even in Taiwan because a fraction of the country don't want an independency, represented mostly by the Koumitang and other part that want an active diplomacy looking for the independence which is the Democratic Party; actually after Ma Ying finished his term, Tsai Ing-wen promised an active diplomacy looking for new trade partners and talked about recognize Taiwan as a state even in the UN.
To do this Taiwan is looking in two ways: the first strategy is the oldest. Taiwan has 28 countries supporting it and each year, to put an example, one of his allies, Paraguay, present a proposition in the General Assembly (UN) to talk about the Taiwan issues. Most countries, which have a strong relation with Continental China rejects this but each year they're still trying.
The second strategy is looking for new commercial partners. As we said before, Taiwan has 28 countries supporting but they're not industrial partners nor developed countries, most of them are developing countries that Taiwan contributes with technical assistance and donate money to keep their votes at UN alive. One of the strategies Ma Ying promised in the electoral campaign to start talking with China about more freedom to access new trade partnerships in the region.
China will make this a nightmare while Taiwan will try to earn some muscles and gain more allies, hoping to add some developed countries with veto power such as UK or France (Continental China and Russia are allies in this case and United States was the country who proposed the recognicemtn of The People Repblic of China as the real China) join with them or pressure some UK and France's major commercial partners to ally with them.
The United States defends its veto of a strongly supported UN resolution that would have paved the way for full United Nations membership for Palestine. The US stresses that while it supports a two-state solution, statehood must be the result of negotiations between Israel and the Palestinians.
There's no way to bypass the UNSC veto as the United States demonstrated when it vetoed Palestine's membership to the UN. As long as China is a permanent member of the Security Council, Taiwan won't be able to join the UN.
