I had to read this a couple of times to understand but here's my opinion.

1 - I completely agree with and would be very useful.

2 - correct me if I am wrong, is already implemented?

3 - I now understand and can reproduce.
You wish for cooktimer to switch on stopwatch mode keeping it's current time, if an item is selected (which currently it resets to zero). However if no item is selected (Choose Dish) stopwatch mode will start as current. Makes sense.