This sounds like your leach field is possibly blocked. That's the disaster scenario where you not only need the tank pumped, but the yard dug up and a new field placed.
Water from your laundry coming up in your tub is a sign that it's not draining to the septic system. This is either because there is a block in the system before the tank or because the tank is unable to accept any more water as it is plugged on the discharge / leach field side.
When I opened up the clean out pipe clear water came out (about ten gallons). I shoved the hose down about 25 feet and it was running full force but it seemed like all the water coming out of the hose was draining back out of the clean out pipe.
... and that's the sign that it's doing option #2: the tank is full and failing to drain.
No, there should not ever be two traps on a single drain line.
The point of the trap is to create a water barrier which prevents sewer gasses from coming up the empty pipe and into the house.
When you have two traps on the same line, you end up creating a kind of vapor block in the line which prevents it from draining properly.
What do you do?
Get yourself some 1 1/2" PVC (not CPVC, which is yellow, PVC which is white) with a 90 deg elbow, and also get two Fernco couplings (black plastic couplings with straps on each end that can be tightened) that will fit the pvc and iron drain pipe.
Use a metal blade on a sawz-all and cut out the p-trap in your basement. (Have a bucket handy!) Leave room to connect the Fernco coupling on both sides. I'd cut the right hand side out just past the coupling.
Measure and cut the PVC so that with the 90 elbow the ends line up with the newly cut drain lines. Glue the PVC together and confirm the fit, then connect to the old drains with the Fernco.
Viola - a proper drain line that will drain smoothly w/ no vapor lock.
Best Answer
There is a plug below the floor drain. This is why when the washer drains it backs up. Soon it will start backing up when you take a bath. The reason it got worse is the land lord knocked more debris into the plug and it is draining even more slowly. A long power snake is the best way to clean these problems as it will get worse with time, drain cleaner may help but I wouldn’t count on it. (I say it will start backing up with a bath, this depends if the bath is on the right side of the plug).