From what I know, the Prophet of the LDS Church holds the position of Seer, as Joseph Smith did. If this is so, what specific privileges and powers come with that position? Does this include the ability to translate ancient languages as it did for Joseph Smith?
The Book of Mormon talks about seership in Mosiah 8:13-18.
Further, President Benson, president of the Quorum of the Twelve Apostles in 1980, spoke about 14 fundamentals to know about following a prophet. In that speech, he declared some of the authority and rights the prophet has while he maintains his office. I've summarized them here but they all demand further explanation and detail (because, I believe, many of these claims are very bold and forward) -- I recommend an in-depth study of the full article and its references.
According to the Book of Mormon:
So we see that a seer is a person who has power and authority from God to reveal that which is not known, including the translation of ancient records, but also knowledge "of things which are past and also of things which are to come," and that this power works through faith.
This is an important point when discussing the hypothetical abilities granted by God to such a person, because in the Book of Mormon we find an explanation of faith that closely parallels Paul's definition from Hebrews 11, but with one important clarification (emphasis added):
So faith is only valid when it is based off of something that is actually true and correct. This means that a seer would only be able to use the gifts that God has granted to him for purposes that conform to God's plan, and not for arbitrary or personal use.