I'd put the planks in as shown on the last photo so that any water-damaged or heat-damaged ones could be easily replaced.

Test the click system by seeing how easily you can snap off the tongue.

Is this in the rest of the house ? If so it ijust continues into the laundry room . Not sure what your situation is .

I agree with running it whatever direction your floor runs in the room it meets up with.

if nothing to match in the doorway, run it the length of the room not across.

Do a dry run, and look for all the variations before you install.

Hide discolored or odd colors under the W/D if you need to.