Is it the proper role of a journalist to interrupt and talk over an interviewee when they don't like the answer the interviewee is giving?

No. They are not only free to challenge an answer, they have a duty to challenge an answer, but they wait until the answer is completed. They do not interrupt, they do not 'talk over', they do not argue. It is true that many interviewees give non-answers or make other attempts at stonewalling questions, but there are proper tactics for dealing with that. Often, the non-answers speak louder than anything else the interviewee says - the viewer or reader can figure it out easily enough. Repeating the question, sometimes to the point that it appears to be badgering, can be an effective tactic. Mike Wallace used it with great success. But I don't recall ever seeing him talk over people or interrupt them.

When you begin to argue, you have lost all objectivity. You have chosen sides, and you are arguing for yours. Or pushing an agenda that you agree with and/or badmouthing a viewpoint you don't agree with. You cease to be a journalist at that point. Call yourself a commentator, pundit or anything other than a journalist.

Edit: I did not see the Kelly/Huckabee segment. I was thinking of the recent incident in Florida where Trump rightly told a hack reporter who interrupted, talked over and argued with him to be quiet.

it is the 'proper' role of a TV interviewer to get a batch of questions answered.

You can be quite rude but polite: "So in what way does what you have just said provide our viewers with an intelligible answer to the question of..... ?"

Commercial channels all push an agenda and sometimes the interviewee turns up to deliberately stymie that agenda.

Absolutely, if the interviewee is not responding to the actual question they were asked, and instead goes off on some barely-related tangent, it is the journalist's OBLIGATION to rein them back in!

Yes

You have this odd idea that an interview should just be a monologue from the person being interviewed.

That contradicts the basic definition of interview.

An interviewer asks questions. When the person who is being interviewed refuses to answer that question, but instead reverts to some memorized, off-topic talking points, it is the responsibility of the interviewer to bring them into line.

Mike Huckabee just wanted to avoid the question, get some free air time, and cash his check.

No. If the journalist later wants to edit the answer, so be it. The journalist should politely hear the person out, then, if the journalist has a point to make himself, try to frame the next question so as to reflect that point.

That is generally considered to be rude unless the interview is of the interrogation type where the interviewee is avoiding giving direct answers like many politicians.

Of the modern age Muppet journalist yes they aren't there to interview as such they are pushing an Agenda.

No shortage of shills on TV they are the sort of people that will sell their own grandmother.

Wasn't she "talking over him" when he refused to give a direct answer to the subject at hand?

A professional journalist would never do that; only hacks and wannabes would.