Student or Learner
"I'm going to put my T-shirts in the wash."
"I'll do the wash later."
What about the washing? "Hang the washing or wash out."?
Hang the washing out.
Hang out the washing.
Neither a teacher nor a native speaker.
In another context, it might go as follows: 'We just can't afford that dinner, honey! All right, all right! I may've to do some washing up for it'. (At a restaurant)
US perspective: When it comes to the laundry, I would use only "the wash" and not "the washing."
I'm not a teacher, but I write for a living. Please don't ask me about 2nd conditionals, but I'm a safe bet for what reads well in (American) English.