I was very much under the impression we do in fact believe in a fire-and-brimstone hell. In Nidchei Yisrael, chapter 21, the Chofetz-Chaim writes about the different types of fire in Gehennom.
Yes and no. There are different levels of soul-cleansing. The highest level of gehennom (in which the innermost parts of the neshama are cleansed) is purely shame. Not shame as we know it, but spiritual infinite shame. The person simply stands before G-d with his actions of rebellion revealed before himself (and H-shem of course). Any type of hell you can imagine is infinitely better than this. This is the absolute most torturous experience which exists within creation.
Additionally, there are lower levels of cleansing which lead up to this, in which the more external blemishes are purged. These levels resemble the fire type, and can even take place here on earth.
Shaar hagilgulim discusses this at length. See also Reishis chochmah shaar hayirah ch. 12.
Many of the seforim which describe the details of the gehennom experience, speak allegorically. So you need to be discerning as to what they really mean. Especially since a spiritual experience simply cannot be described by human language, and only hinted at to those who already are aware of the type of experience, in a way of referral as opposed to explanation.