Up sh*t creek...

I'm sorry but I tried to help earlier but had problems getting in the site. Take a garden hose and feed it into the floor drain. If you can get it through the trap great, if not it will still work. Pack an old wet towel around the hose and floor drain and holding it tight have someone turn the water on. If it's a clog this will usually clear it. If it doesn't it could be roots or a broken tile or fitting. If you have a junk hose that you can cut the male end off of you can usually feed it all the way out to the city main to clean the line. If you have an older home the sewer tile are probably clay and roots can get inside easily. If a newer home with PVC sewer you should be good with a cleaning. Your floor drain is cast iron and the plug is ductile iron and the two usually don't rust together badly. Do what Rob said and PB Blaster it over night.
:wack:
I can tell, there`s not a plumber in the bunch! okc was fairly close, rrr was in there a little too, most of the other ones were shots in the dark. first off, it does sound like a mostly plugged drain, not a vent problem. second, you can`t go down thru ur washing machine drain-as there is a ''should be 2" p-trap at the bottom of the stand pipe. if ur floor drain is cast iron, the plug should be brass, not ductile iron, it could even be pvc if it has been changedby some one. does ur house have studor vents in the attic? if not there should be more than one vent exiting the roof, "if" ur grand pa or who ever plumbed it new what he was doing, unless all the plumbing is bunched up real close together and not scatered all over the house(a lot of distance between fixtures). you can take a small torch and heat the clean-out plug inthe floor drain(get it hot!), that way u won`t be trying to shove ur snake -garden hose or what ever u use thru the floor drain trap. and it is supposed to be hooked in w/ the house plumbing! if that doesn`t work, ur going to have to find the spot that the trunk line exits the house. dig it up, make a hole in it drill-it or whatever(depends on the mat`l ) if it is clay tile u can use a masonary blade carefully to make a small hloe, of the old-old fashoned way and chipp it very slowly a bit at a time w/ a sharp little chisel an a medium size hammer. careful, u don`t want to break that pipe! use ur snake both ways, in toward the house and out toward the yard---u still might need an auger type sewer machine!--if u have to u can bend some scap metal that won`t rust quickly and fashion a cover for it to keep dirt out of it, obviously a clean out tee would be better!---good luck-----bob