The Silk road was essentially a series of trade routes, the exact locations of which varied over time, but generally connected the same destinations. some of which were caravan paths, some were just general routes as you said. The actual paths were not at all like Roman Roads. Often they were barely worked caravan paths.

If you look at a map of the land routes silk road (there were maritime routes but that's not exactly relevant to your question), there is one major "artery" so to speak, from china to the east Mediterranean. At different times, different "tributaries" went from the major silk road to large cities that were not on the main route. Under Mongol Rule, Karakorum was connected, and in the first century, a route connecting northern cities in Indian (Mathura for example), existed. While these may not have all been the same pathways, there was a general route, 'decided' both by geography and city placement.

As for the second question, the answer is 'it depends'. During chinese control, the bureaucracy generally enforced a tax policy. When the mongols controlled a majority of the silk road, they imposed taxes and protected the road, and preferential pricing for higher up mongols, or the sale of goods within the mongol territory, or the exportation of goods (essentially using mongol cities and posts as trade hubs) could result in a smaller tax along with other benefits. In the medieval era, Turkic tribes such as the Magyars controlled the silk road, and it was obviously less unified, so to the extent that Turkic nomads taxed the traders, it was rather inconsistent. However, the silk road led to the settling of nomadic barbarians and pastoral tribes, attracted by all the money to be earned. A lot of times, tribes would invade and occupy cities on the road, forging impressive empires. Again, tax policy among these newly settled peoples was inconsistent.

Did Columbus discovering America eventually change the road? Did it eventually shut it down?