The other answers are correct in that mp is always meant to be louder than p; that's simply the meaning of the term.
Regarding the Rossini specifically, it's interesting to note that he originally wrote the sonatas (at twelve years old, no less!) for two violins, cello, and double bass. Only later was it rearranged for a traditional string quartet and published. Additionally, it doesn't seem to be clear whether he did this later arrangement himself or not. He was known to have disliked the piece a great deal, and might have had nothing to do with it.
So it's entirely possible that those dynamic markings are not related to the original at all, and were instead an attempt by the editor to improve the piece - the only (later) edition I've seen doesn't contain anything of the sort, rather the movement is marked p as a whole with the occasional sf. If later editions follow the original manuscript (which was rediscovered, I believe, in the 1940s), it's possible that they are more accurate to the original, for whatever it's worth.
Still, either way I wouldn't throw out the possibility that it's meant precisely as it's written: piano, decrescendo, then immediately to mezzo-piano. That sort of indication isn't the most common but I've certainly seen it, especially in concert band music.