There is a possibility, when you select a contour line and give a fixed length of line, this line connects automatically to the next contour lines? I have two other questions on this subject but I do not have an adequate response. This is very important for the design of forest roads, to select an optimum zero line, depending on the longitudinal inclination of the forest road. (1-12%). Qgis 2.14.1 Essen
Using this tool you can specify a distance. Then you just need to set up your snapping options to snap on intersection with your contour layer. As you can see in the image below a dark blue circumference guide helps you visualize the extent of your reach.
If I understand your question, you want to design a road thru a sloped terrain, with start and end points. And you need to keep the slope of the road at minimum. If this is the case, then working with contour lines just complicates the process. There are "least cost path" tools available that work on raster layers. So you should revert your contour lines back to an elevation raster (from the grey scale pixels above, it looks like you already have an elevation raster) then, using i.e. GRASS GIS modules r.cost and r.drain you can create your minimum slope road like so:
# First create an elevation raster, if you don't have it v.to.rast <your_contour_lines> output=contour_rast r.surf.contour contour_rast output=elev # Use the elevation to create a cost surface, then find least cost path r.cost elev output=elev_cost start_coord=<your_start_x,your_start_y> stop_coord=<your_stop_x,your_stop_y> outdir=dir r.drain -d elev_cost direction=dir output=road # Convert raster output to line vector r.thin road output=road_thin r.to.vect road_thin type=line output=roads