I imprisoned my wife: if I exile her, will we be still married?

Yes.

The marriage lasts until death do you part. So either you encourage the death part, or if you are good friends with the Pope (or your preferred religious head), you can ask for him to annul the marriage ... however since you are Cathar, that is not an option.

This is a historically correct depiction of royal marriages in medieval Europe. Lots of regents lived apart from their wives for various reasons. To put it in in-game terms, the good Henry VIII of England installed an anti-pope, because the Catholic Pope wouldn't grant him more divorces.

However, a spouse in exile should lower her plot power, and definitely the fertility rate. (I have no exact numbers, please educate me). Since she is also your lover (sigh), I guess there is still the chance of producing babies, however.

Note, that while she is imprisoned, she can't participate in any plots.

As a sidenote, if she's such a fantastic schemer, and she likes killing of your spymasters ... why don't you install her? Obviously she wants the job. :)

She will remain your wife.

I would not recommend exiling her since that puts her beyond your easy reach. Either hope she dies in jail (she'll probably complain after a bit giving you the option of dumping her into even worse conditions until she croaks) or start a plot to assassinate her.

Execution will bring penalties. Killing her by plot is risky if discovered but if you can get enough plot power the risk will be small.