Terminology: The choke is normally open for riding. It is closed for cold starting. Some say the choke is "on" (closed) or "off" (open).
If you need the choke closed to get it to run, you may have carb problems or vacuum leaks, especially if it runs better as you open the throttle.
The sputtering/dying after riding for a time could be fuel starvation. Since it won't even start now are you sure it's getting fuel?
It's also possible that leaving the key on damaged one of the coils and it fails as it warms up.