We did a load of wash recently and found that the 4 in. floor drain where the washing machine empties had overflowed. Coincidentally, we also found that there was some back up in the hall bathtub (didn't see water, but saw some leftover sediment). Obviously, we have a clog somewhere. But, where? Should we just snake the floor drain line where the washer empties? Can we use a 50ft and 1/2in. Pipe snake?

A cleanout was installed in the frontyard two years ago, right before we moved in. Do we need to snake from there? Will this require something more powerful than a 50ft and 1/2in. Pipe snake?

DO NOT snake backward against the flow from the front cleanout into the house. ALWAYS snake with the flow. That said, you have a blockage down stream from the washer. If the front cleanout has standing water then snake to the street from there. If it's clear then then you can snake from the floor drain, the washer roof vent or the lavatory roof vent that vents the bathroom group the tub's in. If you snake from the roof put out enough cable to reach the base and 20 feet more to get the clog.
A 50 foot snake might not be long enough if you snake from the roof. Good luck, Tom

Tried to snake from floor drain and shower drain with 50ft 1/2 in manual snake, but to no avail. Ran into problem about 2 feet down in both cases. Either I can't make a turn or the line has some sort of trap. Drains are filled, can't see what's in the way. My guess is that there is a root problem further down the line, but can't understand why I can't get farther than a couple of feet. I am tempted to try and bust through, but don't want to damage clay line.

You're going to have to clear the line, clay or not so force the snake on through. If your snake isn't long enough rent a power snake and go down the washer vent. In the meantime let me help with the roots. Going in, let me say that sooner or later you're going to have to bite the bullet and replace that old sewer line with PVC.
RootX or Robics Foaming Root Killer that contain Dichlobrnic are two products you could try.
A less expensive way would be to call around to garden supply stores and ask for fine grain Copper Sulphate. Put a 1/2 pound in your toilet and flush it down. Repeat in 6 months.
Hope this helps Tom