If that was my house, I would end up jackhammering up all of the copper supply lines coming out of the slab, or capping them off and abandoning them. Copper supply lines can develop pinholes over time if directly exposed to concrete and create a leak inside of the slab. Leaks like this are not always discovered quickly, and the repairs can be costly. There could actually be a leak now without even knowing about it. Sometimes the only way to tell is that the water bill suddenly becomes much higher than usual. Depending on your locale, direct bury of copper supply lines is also against code.
It appears that all of this work was not originally done by a licensed plumber. A licensed plumber would not have split off the supply from the main sink for the bar sink. They also would not be allowed to install saddle valves. There appears to be 4 in total. I am not sure what they are going to. I am guessing one is for the refrigerator, or a reverse osmosis system, but I am not sure. The drain is also running uphill which would cause the sink to become clogged on a regular basis. There is evidence of a leak there because the wall is stained and has mold on it.
The first thing to do is to shut the water off to that entire area. Find where the supply originally goes into the slab, and then shut the water off there. If no shutoff exists, then you will need to install one. Cut the pipe off and either cap it, or break open the slab where the pipe is and remove it.
Since you have a slab, I would recommend running the new supply lines above the ceiling and down the wall. Since the supply is going to be on an outside wall, I recommend using PEX instead of copper due to the freeze risk. Instead of branching off of those supplies, I recommend running separate supplies for the bar sink, fridge, dishwasher, refrigerator, etc.
The next step would be to address the drain. Remove the drywall immediately surrounding the drain. You want to expose the stack in the wall and make sure that it is vented. The drain for the dishwasher is not in line with the sink drain, so I am not sure what is going on inside of the wall. Assuming that it is all one drain, cut the stack right above the bottom wye, and above the upper wye and remove the piece. Install a new wye and use a piece of pipe and a coupler to reconnect the vent to the stack. When doing this, you want to check the height of the inlet to make sure that it is in the correct position for the rest of the plumbing. You don't want it so high that there will be water in the bottom of your sink, and you wouldn't want it so low that you could not install the trap without it hitting the bottom of the cabinet. Once that is in place, then you can install the trap adapter to the wye. The adapter has a compression fitting and allows you to completely unhook the trap for cleaning. You would not want to glue it in like it is now.
You can run the drain directly into the garbage disposal. In order to prevent the sink from backing up into the dishwasher, you can attach the middle of the line directly under your countertop. This is often called a high loop. The water from the sink will partially go down the line, but stop before it siphons into the dishwasher.
The rest of the plumbing would be done after the cabinets and sink has been installed. Before the rest of the plumbing is installed, you will want to block off the drain with a test plug, or a rag. If you use a rag, be careful to not allow it to fall down the drain. It would be nearly impossible to retrieve it.
Best Answer
No problem and you can go very much lower. It's done everyday for garbage disposals attached to deep sinks and even laundry tubs. I've even seen traps resting on the room's floor with the cabinet bottom greatly notched out for servicing and replacement (plain ugly)...the sink drained exactly like it should and I had no problems washing my tools and buckets.
Additionally, the sink's tailpiece that goes into the trap assembly can be up to a maximum of 24" long. Initially, I couldn't find a public Code Reference, but there is 1 here https://www.law.cornell.edu/cfr/text/24/3280.606 to substantiate this very rarely seen setup.
Your situation seems to be a basement or on-grade slab. Though, if there were a readily accessible crawlspace or garage below the deep sink and its garbage disposal, an up-to 24" tailpiece could place the trap in those areas and not in the sink's cabinet or room at all. In operation and regulation, there's little difference between a sink, shower or laundry standpipe trap.
Side with immediate and easy access in all cases. Not seeing a P-trap in the sink's cabinet will have most Home Inspectors and Newbie Building Code Inspectors hyperventilating and clutching for their red pen. Although, these idiots will still happily approve all illegal horizontal drywall installations, amazingly.