We have a front loader and yes, once you push the button you have to wait for it to finish.
Just plug it back in, select another wash cycle and let it finish, then the door will unlock. Even though you broke the latch, you should still be able to open it once it unlocks.
As for using detergents, I used to abseil years ago, the ropes used to get really dirty and it was a no no to use any sort of detergent because it will affect the overall strength of the rope.
Just because it's ok to use it on clothes doesn't mean it's ok to use on something else.
You might be lucky this time, but I wouldn't do it again.