If it's a cauldron than he's probably a warlock or at least that's what a lot of people would have said before Harry Potter. I recall a time when the overwhelming majority of English-speakers considered warlocks to be male witches, with both words having a very negative stigma. There are a lot of tales of witches and warlocks dealing with dark powers, including demons. As the Wiccan faith has gained in popularity, and popular television shows like Charmed break the old stereotype of wart-covered evil hags, there has been a great deal of effort spent on reclaiming the word witch. Ironically enough, many of those laboring hard to reclaim the word witch also strongly oppose rehabilitating the word warlock.
Wizards aside, in medieval times it was fairly rare for women to attain respected positions at court that didn't involve being someone important's wife or mother. Except for in certain very rare instances, that door was barred to them. In a more egalitarian world where women can be marshals, knights, castellans, stewards, chamberlains, justicars, sages, scholars, heralds etc.. there is no reason why they can't be wizards too. Heck, even in a world that isn't fully egalitarian, but where magic is a very real and potent force, it would be patently stupid for a king to turn down help from the most powerful wizard around, even if that wizard is a woman.
The word wizard also implies a high degree of formal education, and a degree of scholarliness that the word witch doesn't. In a world modeled after our own where women didn't have the same access to a high degree of education as men, it shouldn't be surprising that few women have the opportunity to become wizards.
Not necessarily. While "wizard" was used almost entirely for male characters and "wizardess" is a mouthful, I did read plenty of tales of "sorceresses" as a child. And while cauldrons are heavily associated with witches, the rest of that sentence could just as easily be associated with a sorceress. In fact, the word "robes" puts me more in mind of wizards and sorcerers/sorceresses than witches.
Well witchcraft has traditionally been tied to "superstitious" peasants who haven't entirely given up their former pagan ways. And when the powers that be have a history of persecuting witches, that gives witches plenty of reason to avoid the castles and towers where those powers that be are most powerful.
Yep, if she can keep a low enough profile to avoid those who would persecute her.