Yes, you can just say "Doushite?" when you want to ask why. To be polite, you can say"Doushite desuka?" I think this is the best and safest way for Japanese learners. You could also say "Naze?" It's often used for formal writing or speech, such as essay and public speech, rather than daily conversation. "Nande?" is another alternative. As you may know, "Nande" originally means "using what, how, by what." It can ALSO mean "Why?"depending on the context. If you use "Nande" in this meaning, it may sound a bit casual and rough, while "Doushite?" sounds relatively soft.

I'm with mami on this. :) (the smiley is for me being just a beginner, LOL) Perhaps the most literal translation of "Doushite" (dou + shite) in English would be "How come?" Like in English, it's like 'why?', but puts slightly more emphasis on the reason for something. Like I could look up to the sky, and utter a single "Doushite!?", to mean something like "How come this is happening to me?!" I think "Naze" is not really suited for that (too formal and lacks the 'how' element). "Nande" is like "Doushite". After all, it's 'nan' (what) + instrument particle 'de', and can therefore also mean just: "How?" ('in what manner?') Like: "Nande itta no?" = How did you go? ('jitensha de,' for example) I think it predominantly just means "Why did you go?", though.

You can say just どうして or just なぜ. I am a Japanese. I use "why" as the meaning of なぜ. And I use "how come" as the meaning of どうして. I am not a bilingual, so I don't know the subtle difference. In Japanese, I use どうしてmore than なぜ　in daily conversation.

I believe so. To be honest I'm not a pro on the subject but I use なぜ to not sound so formal.