Why isn't "hoeveel vogels komen uit de hoed"? translated as "How many birds come from the hat?" an acceptable answer? Is it different when a person comes from someplace than a bird? (Wij kommen uit Nederland=We are from the Netherlands)
I suppose here the birds literally come out of the hat. Unless the birds reside in the hat I don't think it is suitable to say "the birds come from the hat", just like in your example it would not be suitable to say "We come out of the Netherlands"
Have a read through this: http://dictionary.cambridge.org/us/grammar/british-grammar/out-or-out-of
In short: out of indicates direction, out (without of) indicates position.
Why is "How many birds do come out of the hat?" wrong. Why do I have to use present progressive and not simple present?
Present simple = How many birds come out of the hat.
The use of do in this sentence would be emphatic (http://dictionary.cambridge.org/grammar/british-grammar/common-verbs/do) and would require the use of wel in the Dutch sentence.
I am no native English speaker; I thought that in questions I have to use "do"? I am confused.
For yes/no questions, you have to use "do" (in some cases, not always). When you have a question word (like "how", "what", "who", etc), you don't.