Best university in Taiwan should be National Taiwan University.

I don't think HSK is accepted for admission to Taiwan university. Taiwan has TOP, a similar HSK-like test. Since HSK uses simplified characters I don't think it's acceptable to Taiwan.

SPM Mandarin is not equivalent to Taiwanese's high school Mandarin, but whether it is acceptable or not you'll have to contact the universities for further information. But I can confirm, to enter Taiwan university, you must be able to write in traditional character.

Our Mandarin standard is high, but not as high as Taiwan and China. They study literature, memorise poems, read history, write essays and give title to their own essays in their exam. Ours is, frankly, much easier if compared to theirs. Our SPM mandarin is probably equivalent to their 初中考. Not sure, but it's no where their 高考。

Yes, unfortunately, you need to write in traditional characters. If you prefer a Mandarin speaking country, then opt for China instead.

Yes, there is a grading system and you can google them. I am not sure whether HSK and TOP are equivalent.

I have never seen the paper, but last I heard the paper isn't tough for Malaysian Chinese. The level is probably PMR level, maybe lower. Here's a sample and this is advanced level, the highest level for TOP. The paper looks easy for me. There are basically four levels of test: basic, easy, intermediate and advanced. I think you can ace the advanced test relatively easily. Malaysians won't find the paper hard. 

But again, do contact the university you intend to go to check whether SPM Mandarin is accepted for admission. If it's accepted, save the trouble!  They might not need a Mandarin proficiency test too, they might accept you without TOP/SPM/UEC, so you might not even need to have a proficiency test, but it certainly is best to have one.