In this instance, "least" is a noun meaning "smallest, lowest, or minimal amount."
Rephrasing the original sentence to demonstrate this meaning gives us: "Its lowest speed is faster than sound." Rephrasing Fumble Fingers's example sentence gives us: "A ton is its lowest possible weight," or, "The smallest weight it could have is a ton."
The rephrasing also helps clarify a more subtle point about the original sentence in question here. "At least" is taken as definitive, precise, and indicative of a specific point in a scale of measurement. So if the scale of measurement is speed, we might say "at least 700 miles per hour," or "at least the speed of sound." We would not usually say, "at least faster than the speed of sound," just as we would not usually say, "at least more than 700 miles per hour." "Faster" and "more than" are not defined points in the scale of measurement, and so they eradicate the functional bottom limit implied by "least." If something moves "at least faster than sound," its minimal speed can be any speed whatsoever as long as it is faster than the speed of sound.
One could argue that it is possible and reasonable to say "at least faster than sound," or in other words, "its slowest speed is nevertheless faster than the speed of sound." And I can't argue that this phrasing is impossible or non-reasonable. What I can argue, however, is that this phrasing is slightly awkward, and inconsistent with the well-accepted and useful intended meaning of "least." I would then say it is preferred for precision, effectiveness, and clarity of expression to use a single, specific point in a scale of measurement when you are referring to "the least."
Demonstrative determiner is not exactly a syntactic category; determiner is a syntactic category, but demonstrative is basically an etymological+semantic grouping consisting of this, that, these, and those.
So although next is a determiner in some cases, it is never a "demonstrative determiner".
Best Answer
In your sentences, next is an adverb.