For example if I draw a oval and then want to paint part of it into different color, how can I connect edge points of my new path with oval?
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What you need to do is this:
Expand the small path, so that its bounds extend over the outline of the oval (you'll need to click 'Close Path' in the inspector to make it a close path):
Duplicate the big oval:
Select the newly duplicate oval and the small shape:
Select Layer › Combine › Intersect:
Select Layer › Paths › Flatten:
Hope it helps :)
You could use the big oval as a mask. Then everything you draw on top of it will be clipped by it. This way the shapes are a bit easier to edit than with booleans. Masking in sketch works pretty much the same as using clipping masks in Photoshop.
(The join command only works on two open paths. It will connect them from their end points. You can cut off a section of a closed path with the scissors tool.)