No, although "unhappy" is a negative prefix + "happy", this is a negation of the adjective. The verb itself is positive still, and tag questions negate the verb.
She doesn't look happy, does she?
She looks unhappy, doesn't she?
If not is negating a finite verb, i.e. one that is connected to and inflected by a subject, then not should come after a helping verb, meaning verb to be, modals and auxiliary verbs be, do and have, for example:
I am not well today.
I have not finished my assignment yet.
Don't you have better things to do?
She was not paying attention.
This is what is normally called negative clause
However, if the verb being negated is nonfinite, which means infinitives with and without to, participles, except for perfect tenses, and -ing gerunds, except for progressive tenses, then not normally comes before the verb it negates:
It is common not to tip severs in some parts of Europe.
The problem was solved by not using the defective part in further projects.
"The Road Not Taken" is a poem by Robert Frost.
I can think of a hundred reasons not to come.
Likewise, it is possible to negate words and phrases other than verbs, normally by placing not right before the word or phrase one wishes to negate. Your first example falls into this category.
Perhaps not surprisingly, he was cautious about committing himself.
He won not only the championship, but also the hearts of thousands of fans.
Best Answer
In the negative, using "too" is wrong. It can only be used with the positive. Typically you should use 'neither', although in conversation it is acceptable to use 'either' as well. For more info see:
What is the difference between "me neither" and "me either"?