Why Apple watch Nike+ is in same price as normal watch

If the watches are different, and the phones are different, then the only thing that appears to remain the same is you.

You don't mention, but have you ruled out the possibility of there being anything about you? For instance, I remember reading that tattoos could prevent the heart monitor obtaining a clear reading, and there was some talk about people with very black skin also having difficulties. Example article here http://www.bbc.co.uk/news/technology-32537030

The other possibility is whether there is something with the tightness with which you wear it?

You are correct on all three counts.

The Workout app will work without your iPhone, at least. Third-party apps might need to be updated to be completely independent. Strava, for instance, won't start without an active iPhone connection, but Runkeeper will.

Both the first generation WATCH and Series 2 support syncing of a single playlist up to either 2GB or 250 songs.