There are several plugins to join [or aka 'weld'] lines, curves, arcs etc into a single bound 'curve' object, these can then be used in processes, like those in some of my Extrude Tools...
My own version is available in the PluginStore.
http://sketchucation.com/pluginstore?pln=TIG-weld
Preselect a set of connected edges/curves, run the 'weld' tool from the Tools menu item, or setup/use a shortcut key-press... and then it should make a welded curve - but remember there can be no intervening 'branch' edges as these will split the curve where they touch it...