Can a carotid artery bruit come and go with time?

Yes. Trick question, but here are my thoughts. A carotid bruit is due to a partial obstruction in the carotid artery. A "sub-clinical" lesion may only give rise to a bruit during periods of intense exercise, when cardiac output is high. In the resting state, a bruit may not exist. At any rate, a carotid bruit in any circumstance should be evaluated.
Maybe. I would simply get a carotid ultrasound to assess the carotid circulation. If there is a lesion that is causing the bruit you will know about it after the ultrasound.