Under copyright law, derivative works of public domain material is under copyright. I would imagine that this is due to the translator adding things such as a translator's preface, footnotes, etc. However, would the translator have copyright protection to the part of his work which was translated from the public domain material? That is, would he own the words of the translation itself even though he did come up with them?

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yes, she or he would. there is no such thing as a 'pure' translation. every human language has idioms and usage conventions which defy simple logic and must be translated using human understanding of the culture into which the material will be translated.

No, because the original material is part of the public domain. I dont really understand why he copyrighted his footnotes, but eh.

Any work in the public domain by legal definition is available to anyone else to re-publish, translated or not, and the new version becomes the copyrighted material of the translator or re-publisher. Therefore just as a new screen adaptation (translation) of "Don Quixote," for example, would be fair game, so would new language translations with prefaces, introductions and all. They're done all the time.

Yes, derivative works of public domain works are considered "copyrightable", although it is a legal anomaly. As a general rule, the exclusive right to create a derivative work belongs to the copyright owner. The owner of a public domain work is, obviously, the public. Therefore, the public has the right to create a derivative work, which should naturally be owned by the public. If someone wants to own copyright on ANYTHING, including derivatives of PD works, they have to add something creative to it. If you take "To be or not to be, that is the question" (a PD excerpt) and make it into "2B or not 2B, that's the question", it would be debatable (among lawyers and jurors) whether your "creativity" is sufficient to claim that portion of the derivative work as your own, given the prior work is not yours.