A no ball is bowled. The striker hits the ball, but in doing so accidentally puts down their wicket. Will the runs be considered or not? And will be the striker be out hit wicket or not?
A batsman cannot be out hit wicket from a no ball. See Law 24.16:
When No ball has been called, neither batsman shall be out under any of the Laws except 33 (Handled the ball), 34 (Hit the ball twice), 37 (Obstructing the field) or 38 (Run out).
As such, if a batsman does break his own wicket after a no ball, nothing in particular happens: the ball is not dead because no wicket has fallen, so any runs scored count as normal.
(This assumes that breaking the wicket was accidental; if it was deliberate, the fielding side may have a case for the batsman being out obstructing the field).