Right now there is no way to Sync an existent table, so you would need to create a new one synced from the beginning.
In order to switch layers in your map, you could just remove the layer for your old table and add the new sync layer. This way, the map itself will not change and you will still have the same visualization ID in case you have shared it already: links will keep working because it's the same map!
If you want to do the same than I explained above, but your map only has a single layer, make sure you add before the new one and after that, you delete the one that you don't want.
Another approach if you are using your table already in lots of layers, is just to edit the SQL code in each of the layers to point to the name of the new Sync layer that you will create.
CartoDB doesn't support JSON files generally, but GeoJSON files. Some JSON which are not very complex/nested could also be imported, but this depend on the structure of the file. Specifically, for Dropbox and Box connectors, it seems that these are not requesting GeoJSON formatted files to the source right now. I'm going to suggest to the CartoDB team checking this possibility to see if there's something that can be improved.