When drain water from one fixture comes back out another fixture, it means your blockage is beyond the point where those drains meet.
You need to get a good look at the topology of your waste plumbing, to get some ideas about where the blockage is. Clothes washer -> bathtub is a common symptom, because the clothes washer drains a lot of water up high (it has a pump) and the tub drain is down low.
Snaking down a drain is really hard, because of all the curves of the P-trap. If you're lucky, you have a clean-out somewhere, like under the house or behind a wall. Otherwise, you can try removing a P-trap to get the snake in, or removing a toilet. Removing a toilet is not very hard, but can be daunting.
We had a similar problem in this house. There were a few clean-outs accessible in the crawlspace, and our landlord used a hand-held snake in them, but didn't have much luck.
They hired a professional plumber. He pulled a toilet out and ran a huge snake down the drain (not one of the handheld units, but a huge, loud, floor-standing model heavy-duty cable). After running all 75' out, he borrowed another snake from his colleague, for 150' of snaking. This is a single-story house, on a 50'-wide lot, so the snake was well in to the street!
He said he pulled out a lot of roots. This is apparently common in older plumbing. They develop a small leak, and nearby plants grow after the nutrient-rich water, and find their way in to the pipe. It took him about 4 hours.
Plumbing is simple. There aren't a lot of rules to follow. Everything works in obvious ways. You don't need a lot of expensive, complicated tools, and if you do need something big, you can rent it. Anyone can fix plumbing.
But it can also be unpleasant. Screw-on connections may be rusted in place - hacksawing is often easier. There's the spectre of old poop and hair. Things may be difficult to reach - in a cabinet behind a sink basin or in a muddy crawlspace full of rat droppings.
If the job takes a few days, and you're unaccustomed to living without plumbing, it can be a trying time. Especially if there's a whole family in the house, and they aren't sympathetic. A pro will get the job done much faster, since they arrive with the right tools, parts, and experience.
There's a lot of instructions out there, on the internet, at the library, at the hardware store. If you decide to get your feet wet (get it? ha ha) then you will be able to find the information you need.
Best Answer
You are lucky there is a vertical pipe. The teapot lid is often at the bottom of the chamber, so it needs to be pumped out for access. As you suspect, this should enable cleaning out the main house trap. The pipe with the flap that is under water is to access the pipe beyond the trap. Either the pipe beyond or the trap can be the cause of the blockage. Dealing with the pipe beyond likely requires professional service with root cutting equipment.
So lets deal with the trap and hope that fixes it. You need to feed a plumber's snake down the pipe and through the trap to break up the blockage. I've also had luck simply jetting out the trap with a common garden hose with maximum water flow. How successful you are depends on the nature of the blockage. It may also require professional service with the proper equipment.
If you do require assistance, enquire about the cost of removing the main trap. It is typically no longer required by building codes and would eliminate a common trouble spot. Sometimes it's an easy job and other times it means digging up your garden.