Can Blowing Your Nose Constantly, Touching and Pushing It Stretch Your Nostrils?

Manipulating your nose to manage the symptoms of a cold will not change the size of your nostrils.  You can change the shape by placing long term continuous force, but you have to be a member of a tribe to do that.

Blowing your nose, rubbing your nose and other forms of nasal manipulation will have no impact upon the size and shape of your nose.  The changes that you describe in your nose following your cold are probably related to swelling from your upper respiratory infection.  Your nose should return to normal as your swelling resolves with the resolution of your cold.

While the shape of your nose can stretch with constant touching and pushing over time, it is very unlikely that this happened in the time of your cold. It would take much longer to do — we're talking much longer than a couple of days and there would have to be constant pressure, even through the night. Your nose may be a little swollen from constantly touching it, but this will subside and return to normal once you've stopped touching your nose so much.