According to notes I have from a class I took, if a question word (e.g., nani,itsu) is present you use only か, otherwise use かどうか. Also, a plain form precedes か and if it is a noun or A2 adjective だ is omitted.
どうか is derived from どう, meaning that an action or event is yet determined. E.g., 君はどうする = What are you going to do? どうしようもない = There is nothing one can do. [bamboo4]