Congressional districts, like counties, municipalities, etc., from the Census Bureau's TIGER download site are already polygons made up of coincident lines. Bottom line is you won't have the adjacency issues you were showing in your OP. They're digitized at 1:100,000 scale, which is more than adequate for your needs. Dealing with topology is overkill, ignore that part (but to answer your question, it applies rules to geodatabase feature classes, so that you can enforce rules that eliminate gaps, overlaps, etc.).
http://www.census.go...tiger-line.html, choose "113th CD" for the 113th Congressional Districts.
XTools (http://www.xtoolspro.com/) can convert polygons to polylines within ArcGIS. It also claims to smooth polylines using either beziers or b-splines; I haven't tried that tool yet, but let me say I like the XTools that I've tried. You probably wouldn't want to smooth the lines that are also state boundaries, so you may need to do some work breaking lines so that you can unselect them for smoothing.