Is Mandarin Chinese in Taiwan can be understood in China, Singapore, and other places?

No worries!
Mandarin Chinese used in “Taiwan” and in “China, Singapore” and so on is just like the same case as English used in “the USA, the Great Britain, Canada, Australia, New Zealand” and so on.
We may have some differentiations in accents, word choices, character writings, but we can fully understand what the others say/write, and communicate with each others undoubtedly.