We have a setup where our washer is in the basement directly below our kitchen sink. From the looks of it they both drain directly into the same sewer/drain line. To the right of the washer is a floor drain. We have had some issues with the water backing up the washer's drain pipe when the washer is either draining or when the kitchen sink is draining (but only when a large amount of water is dumped in down the sink and then the garbage disposal is run to drain it more quickly. A plumber has been out twice to auger the line, but it doesn't seem to alleviate the problem completely. Could it be something other than the drain line?
I take it nothing backs up the floor drain?
I am going to assume this is an older house. The floor drain is probably not hooked up to the sewer line and drains outside to a rock pit or something which is why it doesn't back up.
Perimeter drain that comes into your house with rain / ground water and feed into your sewer service line inside the house would be odd. Modern setups have perimeter drain go to a sump well and the sump well then outflow to the storm drain. In the case where the city doesn't have a storm/sewer main separation then these typically Y right before the entry into the sewer main. Ideally you don't want them connected as if the sewer backs up it would go into your perimeter drain.
How far out did the plumber auger the line? Did they use a root grinding attachment? 30 feet isn't very far, I can't imagine that takes you to the city main. Even 60 feet typically doesn't cut it.
I suspect your main sewer service line probably has a partial blockage and prevents a surge of water from draining fast enough to the sewer main. Last time I had this the plumber dumped a bottle of knockout plus down it.
I suppose venting is a possibility if the vertical pipe that is receiving the kitchen and washing machine outflow isn't vented the water could struggle to flow quickly enough. When you look on the roof above the wall where the vertical pipe is do you not see a vent? Do you have sewage smells coming into the house, if there was no vent your traps would be siphoned routinely and allow sewer gas to enter your house. Is the kitchen an add on or original to the house?
If the water is backing up the pipe that receives the washer outflow and nothing is coming up the floor drain and that floor drain is connected to the same sewer line then it sounds like your restriction is before the floor drain. What diameter pipe is the vertical piece that receives the washer and sink outflow?