When drain water from one fixture comes back out another fixture, it means your blockage is beyond the point where those drains meet.
You need to get a good look at the topology of your waste plumbing, to get some ideas about where the blockage is. Clothes washer -> bathtub is a common symptom, because the clothes washer drains a lot of water up high (it has a pump) and the tub drain is down low.
Snaking down a drain is really hard, because of all the curves of the P-trap. If you're lucky, you have a clean-out somewhere, like under the house or behind a wall. Otherwise, you can try removing a P-trap to get the snake in, or removing a toilet. Removing a toilet is not very hard, but can be daunting.
We had a similar problem in this house. There were a few clean-outs accessible in the crawlspace, and our landlord used a hand-held snake in them, but didn't have much luck.
They hired a professional plumber. He pulled a toilet out and ran a huge snake down the drain (not one of the handheld units, but a huge, loud, floor-standing model heavy-duty cable). After running all 75' out, he borrowed another snake from his colleague, for 150' of snaking. This is a single-story house, on a 50'-wide lot, so the snake was well in to the street!
He said he pulled out a lot of roots. This is apparently common in older plumbing. They develop a small leak, and nearby plants grow after the nutrient-rich water, and find their way in to the pipe. It took him about 4 hours.
Plumbing is simple. There aren't a lot of rules to follow. Everything works in obvious ways. You don't need a lot of expensive, complicated tools, and if you do need something big, you can rent it. Anyone can fix plumbing.
But it can also be unpleasant. Screw-on connections may be rusted in place - hacksawing is often easier. There's the spectre of old poop and hair. Things may be difficult to reach - in a cabinet behind a sink basin or in a muddy crawlspace full of rat droppings.
If the job takes a few days, and you're unaccustomed to living without plumbing, it can be a trying time. Especially if there's a whole family in the house, and they aren't sympathetic. A pro will get the job done much faster, since they arrive with the right tools, parts, and experience.
There's a lot of instructions out there, on the internet, at the library, at the hardware store. If you decide to get your feet wet (get it? ha ha) then you will be able to find the information you need.
I disagree with BMitch as far as if the P-trap will come apart. (Possibly the first time we've disagreed?) The metal nut could not be where it is if the joint did not separate. The rest of his answer I agree is the easiest fix. However, if possible, physically cleaning the trap is preferable to a chemical approach. I'm assuming there is a slip fitting just out of the picture, just as there is in non-disposal side. By detaching the metal ring in the first photo and the unseen slip joint, the trap should be able to be removed.
Even without a slip joint, the pipe could be detached right at the disposal itself. This could actually be preferable as it would allow the P-trap to swivel, which may help in loosening the nut. Are you sure you are trying to turn the nut the correct direction? I'm not saying you don't know how to loosen fasteners. The direction depends on which piece the nut is threaded onto, which is not always clear in plumbing.
The curved shoulder of the nut tells me it is threaded onto the U-bend, so the nut is turned anti-clockwise when looking down at it. It's hard to tell by photos, but you appear to have decent access to the nut from the left side as it is below the pipe coming in from the other side. Even though the opposite trap may be in the way for fully turning a wrench, you seem to have enough room to place the wrench handle near the trap and pull right.
As for counter force, if pulling on the pipe itself isn't working, try a large screwdriver wedged in the inside of the U-bend. If you have the right wrench, the ring nut shouldn't offer a lot of resistance.
Best Answer
I'm not sure anyone has answered your question yet. It's hard to tell from the picture but I'll defer judgement to the "engineer" on site that the drain line from the dishwasher is somewhat lower than the drain feed out to the sewer line. If that's the case this would cause some of the water (weir) in the trap to back up into the lower part of the dishwasher drain.
To put this in perspective - it's probably not going to cause you major issues. However, since it's under warranty - have them fix it. I'd recommend isherwood's solution above - have them shorten the leg above the dishwasher drain and the corresponding ell going out to the mainline.
The reason I recommend that is based on the picture. The vertical rise from the trap to the horizontal drain piece looks to be border-line long. To explain, here's an illustration of a proper setup:
The height of the weir (area between the red lines) should not be more than 4 inches. Again it's hard to tell but the weir on your trap looks higher than 4 inches which would make it less effective in flushing out debris.
Edit
If the weir height is no more than 4 inches the installers have the option of moving the dishwasher drain up to the higher unused fitting going into the tailpipe on the right or raising the trap as indicated by thye blue lines in the picture. As Harper mentioned in his comment, those pipes may be pre-fab and not adjustable so the installer may have to install new pipes above the trap to get the dishwasher drain line above the outlet pipe on the left.