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
Shut the water off to the house, drain the water from the copper lines, cut them below the Ts leaving enough for a sharkbite to grab back on, put sharkbite couplings on. For the pvc drain line I'd flush cut that to the fitting with the green line and get a furnco with a worm drive to attach them back together.
You can then remove the sink and all supply lines and drain lines when you need to remove the cabinet. You should get the sharkbite disconnect tool.
Un-sweating copper under a sink cabinet isn't a job I'd sign up for - cutting seems easier. After you have your new sink and new cabinet you'll probably want to go with new shutoffs and supplies anyway so I wouldn't be looking to keep that portion of copper. I'd convert to pex and go with 1/4 turn ball valve shutoffs - the shutoffs in your picture I find stop working, leak when turning and are harder to use. You might as well just replace this portion of plumbing the last thing you want after you get a new sink / cabinet is for old plumbing to leak and ruin it.
Shark bite fittings have an integral o ring which probably has a shorter lifespan than a sweat fitting - I only use them in exposed plumbing applications. A lot of plumbers actually use them all the time even in concealed plumbing because they are a 100 times easier than soldering.
Furncos are used in commerical plumbing all the time - this is how you join cast iron (quiet pipes, etc). They are made of some kind of flexible rubber like material again the rubber isn't likely to have the same lifespan as the cast but in exposed plumbing and under the sink I wouldn't be afraid to leave it.
When you do your final install you could leave out the furncos and sharkbites if it makes you feel better. They are more expensive than standard fittings and maybe you have another job waiting to use them.