"Whose computer is this?"
这电脑是谁的 is also correct. 这是谁的电脑 is textbook chinese whereas the former is more natural chinese
Normally you'd need a measure word, but you can say "这是..."or "那是..." when saying "this is..." or "that is..."
I never know where to place the 谁 in a sentence. It changes around quite a lot, any tips?
Whereas English is a language that puts its wh-words in the front of the sentence in questions (as in, "Who is that man?"), Mandarin keeps the wh-word in the position the noun would be in if it wasn't a question. For example, in English it'd be "Who is that man?", but in Mandarin it'd be "那个男人是谁？" (literally: "That man is who?").