Nereye = (to) where, whither -- indicates a direction

Nerede = where -- indicates a place

"Nerde" is how most people pronounce "nerede", but you shouldn't write it like that. (Just like you shouldn't write "wanna" - fine in speech but not (yet) appropriate in writing.)

Those are for yes-no questions and "multiple-choice" questions, but you don't need them if you have a WH- word such as nereye "whither? where (to)?".

It is fine, just a missing alternative.

Yes, it's koşmak, so you take the -İyor- ending (capital -İ- standing for a vowel subject to four-way vowel harmony) and it becomes koş-uyor-sun with the -İyor- turning into -uyor- after the back rounded vowel -o- of koş-.