1. Are "the" and "a" counted as words?
Yes! They are counted as words in the writing, listening and reading tests.
2. How can I improve my listening and reading?
The simple answer is: do more listening and reading. There are no shortcuts I'm afraid; it takes time to develop a "good ear" for a language, and the reading exam is basically a test of your vocabulary knowledge. Your listening and reading will improve gradually with time, practice and exposure to the language.
3. If the question asks for "type of exam", would the answer be "open book" or "open book exam"?
You don't need the word "exam" in your answer because it is part of the question. However, I don't think "open book exam" would be marked wrong (if 3 words are allowed).
4. If the answer in a listening test is "wide variety", would "variety" be marked wrong?
Probably. I've seen many examples in the listening test where one word is not enough. For example, if the answer is "early morning", students who only write "morning" are marked wrong. If a second word changes the meaning a little (e.g. early morning, wide variety), you should write it.