Are translations of the book of Mormon into other languages made from English, or are the golden plates stashed in a temple somewhere?

Of necessity, all versions of the original Book of Mormon, and all translations of it into languages other than the one Joseph Smith produced in English, have to be done from the oldest available English papers kept by LDS authorities. That is because the one and only set of plates (bound by rings) and written in some form of Egyptian hieroglyphics, was apparently handed back to an angel, by Joseph Smith, after he had produced those English papers. The LDS church reports Joseph Smith’s own account of this:

“When, according to arrangements, the messenger called for them, I delivered them up to him; and he has them in his charge until this day, being the second day of May, one thousand eight hundred and thirty-eight.” Joseph Smith – History 1:50-53, heading ‘Joseph’s First Visit to the Hill Cumorah https://www.churchofjesuschrist.org/manual/the-pearl-of-great-price-student-manual/joseph-smith-history?lang=eng

It then becomes vitally important to know how Smith could have produced an English-language translation from hieroglyphics that he could never have known how to read or understand. The Book of Mormon 9:32 says this language was called “Reformed Egyptian”. Surprisingly, the LDS account shows that the plates were to one side while Smith put ‘seer-stones’ into a hat, then stuck his head into the hat to read what the seer-stones said, and to write what those English words were.

An LDS answer to a related question states:

"Joseph wasn't given the ability to read and understand the plates, and then carry out a translation with this knowledge. He was given the translation directly.” Why were the golden plates necessary if Joseph Smith received his translation of the Book of Mormon by viewing a stone in his hat?

How so? The LDS church explains that objects called the Urim and Thummim were used by Smith. Back in Smith’s day they were more commonly called ‘seer-stones’. The golden plates were in the room with Joseph Smith and his wife during the process of translating. Under the heading 'The Mechanics of Translation' this LDS document says:

“Some accounts indicate that Joseph studied the characters on the plates. Most of the accounts speak of Joseph’s use of the Urim and Thummim (either the interpreters or the seer stone), and many accounts refer to his use of a single stone.” Source: lds.org/topics/book-of-mormon-translation?lang=eng

So, what need to have any plates in any unknown language or unknown hieroglyphics if knowledge of that isn't needed to translate? Joseph Smith didn’t need the plates to translate into English. He did need ‘seer-stones’ (and apparently a hat, as well). Therefore, once that first English translation was done, the plates could be handed back to an angel, for there was no further need for them. From then on, copies, and then translations into other languages would be produced from Smith’s original work.

Finally, your last question, “If foreign translations can be made from the English, why not a NIV-style English translation from the English?” Everyone notices that the original Book of Mormon in English uses thousands of phrases and words already written in the Authorised Version (first produced in 1611). This means that the LDS church is duty-bound to stick to that kind of language – Elizabethan English. If they tried to produce an NIV-type version, they would begin to wander away from the wording that apparently had been used by the people in the Book of Mormon hundreds of years before the A.V. was translated.

There is a specific reason that the Book of Mormon is not translated into more modern English.

From a 1993 statement from the First Presidency of the Church,

From time to time there are those who wish to rewrite the Book of Mormon into familiar or modern English. We discourage this type of publication and call attention to the fact that the Book of Mormon was translated "by the gift and power of God," who has declared that "it is true." (Book of Mormon title page; D&C 17:6.) The Prophet Joseph Smith said that the Book of Mormon was "the most correct of any book on earth." (History of the Church, 4:461.) It contains "the fullness of the gospel of Jesus Christ." (D&C 20:9.)

When a sacred text is translated into another language or rewritten into more familiar language, there are substantial risks that this process may introduce doctrinal errors or obscure evidence of its ancient origin. To guard against these risks, the First Presidency and Council of the Twelve give close personal supervision to the translation of scriptures from English into other languages and have not authorized efforts to express the doctrinal content of the Book of Mormon in familiar or modern English. (These concerns do not pertain to publications by the Church for children, such as the Book of Mormon Reader.)

You're right, the golden plates were returned to the angel Moroni at the conclusion of the original translation into English.

First, we have to understand the translation process (this link is the source to the quotes below).

There are some original translation manuscripts still in existence, but these are hardly legible anymore and, for several reasons (such as completing the whole translation in 60 days), contain many grammatical and spelling errors. Subsequent revisions of the English text referred back to the original manuscripts and other historical documents to fix technical issues and improve formatting. These have been published as new editions (most recently in 2013).

Regarding the relationship between textual revisions and the correctness of the translation:

"All who have translated are keenly aware that it is a rare translation which cannot be improved,” said Stephen Ricks, a professor of Hebrew at Brigham Young University. “Thus, while it would be incorrect to minimize the divine element in the process of translation of the Book of Mormon, it would also be misleading and potentially hazardous to deny the human factor."

With an English translation that is close the original writ, translations into other languages are based on the current English revision. Several other languages, such as Spanish, have been revised several times for the same reasons. Some translations are even incomplete and are only available as selected excerpts until a full translation can be made. (I know someone who is helping to re-translate the Book of Mormon into Persian. She told me that the current Persian edition has trouble with some of the word choice and sentence structure compared to the English reading of it.)

As you might imagine, these translations are time consuming and difficult to do well. The translation process is done by study, prayer, and faith.

Finally, to answer your last question ("Why not a NIV-style English translation of the Book of Mormon?"), I would have to say that I don't know. I can't find any Church statement on that question specifically, but maybe somebody else could find an official source.

Well, Jacob Glad's answer added the missing piece, an official statement on the matter. Anyway, my original thoughts were:

If you want my guess, though, it's because the LDS Church wants to maintain the purity of the translation as much as possible. Translating into a different style of English may cause confusion, since it's quite likely elements of the text would inadvertently lose or change the meaning. As it stands, the language found in the current edition of the Book of Mormon is not nearly as archaic as the language in the KJV Bible, and is (in my opinion) still much more readable. With that perspective, some scholars suggest that an official, NIV-style edition of the Book of Mormon is simply not necessary.