"Whose computer is this?"
Colloquially it's sometimes left out, at least in China. Discussion:
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?").
Yes. Both are technically correct, but "shei" is the version that's actually most commonly used. Some dictionaries list "shui" as the main pronunciation, but in my experience, that pronunciation is virtually completely unused, and I would call it almost completely obsolete. However, it's something that seems to hang on due to education (inside and outside of China) not keeping up with actual usage. You may hear it in formal reading, as well as in language teaching programs such as Pimsleur (where I heard it last).