I'm editing an image using Inkscape like this:
Since I have painted image, how can I cut out all the extra strokes (actually whole image are path, including strokes) and creating a new path with only necessary part like this:
One method is to draw a secondary path to cut another in two.
In Inkscape use the Bezier tool to draw a path where you want to cut it. Select both paths, and click Path > Cut Path
Another method is to fill the shapes with white, and then change the order in the stack by using Object > Raise/Lower, etc. There are also buttons for these operations on the tool bar in the Move tool options along the top.