1) As I know, 'quite' has 2 meanings which contradict each other. One is 'fully',completely', and the other one is 'rather','pretty'. How am I supposed to know which one is meant?

2) 'Put (an) emphasis on (smth)'

When do I need an 'an' article (if I need it at all)?

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Comments  (Page 2) 
"quite" is not used before "not", just "after", if at all, so 1 and 3 are not possible. You have to substitute (for 1), "I completely misunderstood", and (for 3), "I completely failed". It is not usual to use "quite" in these: ?"I quite misunderstood", ?"I quite failed", but you might hear these once in a great while. Even my contribution "I quite understood" probably should have had a question mark on it. "quite" before a verb is not as common as "quite" before an adjective.