Totally agree with extra cleansing if you haven't done that yet.
A thought for you since you're so overwhelmed: do you have a little used bathroom or similar small, easy to clean, relatively puppy proofed room, that you could turn into a puppy pen for now? If not, do you have access to a wire "exercise pen" (I actually use a plastic PlayYard XT designed for kids)? With either option, you would layer the floor with papers or puppy pads (use a plastic drop cloth or old shower curtain under the papers in the pen if you don't have a room without carpeting), put puppy's crate inside, some toys, water bowl, etc. Set up the pen so that food/water and bed are all in one area and there's a large open space for moving about.
You can use this room/pen instead of the crate to keep him safe from household dangers - it will not keep him from pottying since he'll have more room, but it will mean that you don't have to worry about peeing in the crate or anywhere unprotected, nipping the kids, etc. for a little while to give yourself a break.
In the long run it should not impede your housetraining if you're consistent about keeping the floor covered with paper/pads to keep him from learning it's ok to pee on carpet, etc. and are careful about supervising when he's in the rest of the house (as you would if he were using the crate, too). It may also pull him out of the peeing in the crate issue since he'll have options available other than that.
eta: (((hugs))) I thought I'd let you know that we all have our moments of uber frustration - I've been training dogs for years but my last puppy (now almost 2yo!) drove me batty quite frequently.
After that experience, I now tell my students that sometimes timeouts are more for the people to regain their composure vs. really teaching the dog something at that moment.
There is no shame in needing a break from your puppy, especially given all that you're dealing with right now - it WILL get better, though, I swear!