Can I use existing telephone wire as a pull for new cables?
It depends. If it was installed originally, it should be stapled, in which case you can't use it. If it was fished through the wall after the fact then yes you should be able to pull using it. 

I would recommend pulling polyline first and then pulling the new cable through, along with another piece of polyline.  

The cable will stretch and break if you pull to hard, and if it goes around hard bends it will very next to impossible to pull all the way through, so you might still end up needing to cut some access holes to use as pull points.