Update on what I have done so far:
I bought a 50 foot endoscope and zip tied it to a 25 foot 3/16 inch auger. I tried to go through the drain pipe at the washing machine but I could not get it to turn the first corner. It just hit the bottom of the 3 1/2 to 4 foot drain and stopped. I did notice that there was standing water in the washer drain pipe even though it had been hours since the washing machine was used. When I turn on the washing machine and let it drain, it obviously pushes the standing water down the drain; otherwise it would overflow almost immediately. Next, I went through the drain underneath the sink. I was able to get the snake to turn the corner at the bottom of the drain, but the snake stopped after about 4 feet. I'm thinking that the snake went right towards the washing machine drain which is about 4 feet from the kitchen sink rather than going towards the main sewer line.
Questions:
1. How do you get the drain snake to turn corners?
2. Is it likely that the drain from the washing machine would flow directly through the same drain as the sink since the sink is 4 feet from the washing machine and along the same back wall?
3. I can rent an industrial drain snake from Home Depot. Do you think this would be a better option to get the snake to turn corners?
4. If the drain pipe at the washing machine has standing water in it hours after the washing machine has been used and the drain under the sink is relatively clean with no water as shown in the pictures, what does that tell us about the drain from the washing machine to the sink? Is it possible that it doesn't use the same drain pipe as the sink? Could there be something between the sink and the washing machine that is stopping the water from completely draining after hours of non use? Could a venting problem cause the water to stand in the washing machine drain?
5. Can anyone look at the pictures I have attached and tell anything from them? I know that all of the pictures where there is no water were made from the drain below the sink. I think the pictures that look wet were made from the drain at the washing machine, but I can't be sure. I need to try everything again and note exactly when the pictures are taken and from which drain.
I believe my endoscope can help me determine exactly what my issue is, if I can get it connected to a drain snake that I am able to get to turn corners and go in the direction I need it to. I have already had a couple of plumbers out and they used drain snakes from the washing machine, below the sink and from the cleanout from the main sewer line outside.
I don't believe any of the local plumbers have an endoscope. I will need to call a plumber from about 1 hour away to get one that has every tool needed to figure out what is going on.
I don't mind spending money trying to figure this stuff out on my own since I will be learning things and I may actually get it resolved with your help.
Images 5 through 10 are when I put the snake down the kitchen sink drain the first time and shows where the snake stopped after about 4 feet. Images 11 and 12 are when I put it down the kitchen drain a second time. I believe image 13 is from the washing machine drain, but I could be mistaken on that one.