Could the silk roads have started before the Han Dynasty?
Absolutely. The Silk Road has a LONG prehistory. According to the Wikipedia, Lapis Lazuli mined in Afghanistan was being traded all the way west to Egypt by the second half of the fourth millennium B.C. In the second millennium China was mining lapis lazuli in nearby Khotan. And of course there are those European Mummies from the same period found in the Chinese Desert. The Han Dynasty got its start in about 200 B.C. That was less than a century after Alexander and about 3 centuries after Darius, both of whom made the area from Greece to India relatively stable and safe for trade. Nobody is sure whether Pasta evolved independently in the east and west, but the truth is that by the second millenium at least it was being eaten both in China and the Meditteranean. There is plenty of evidence to suggest that yes, the Silk Road started well before the Han (or the Qin for that matter) dynasties although it really got up to snuff when the West could afford silk.