I was not able to google even a single instance of out "Hit the ball twice" in cricket.
You didn't try very hard! The Wikipedia page for "Hit the ball twice" lists two examples.
Can a batsman be out "Hit the ball twice of a no-ball"?
Yes. You yourself linked to the offical Laws of cricket which state:
- When No ball has been called, neither batsman shall be out under any of the Laws except [...] 34 (Hit the ball twice)
That seems about as clear as it could be written.
How many runs can be scored of the no-ball?
One run will be scored for the no ball. I'm struggling to imagine a realistic situation in which the batsmen can complete a run before the striker hits the ball for a second time - note that once a fielder has touched the ball, a batsman cannot be out hit the ball twice. Law 37 (Obstructing the field) explicitly includes the fact that runs made before the dismissal will count, but there is no such mention in Law 34, so I think this is somewhat undefined in the Laws.
Next ball should ideally be a free hit?
Free hits are not part of the Laws of Cricket, but instead part of the playing conditions for various tournaments, so it is not possible to give a generic answer to this question. However, a dismissal does not stop the free hit from occurring under the current ODI or T20 International playing conditions.
Has it ever happened in the history of cricket?
In the history of cricket, probably, yes in some club match somewhere. I am not aware of any such incidents in professional cricket history though.