That isn't how gravity works.
Drains like sinks are typically gravity drains where the water is drawn down into the drainpipe... by gravity.
This is in contrast to things like washing-machines where the wastewater is pumped out into standpipes.
In order to drain your sink via a standpipe above the fixture you'll need a drain pump or drain into a sump and pump it out from there.
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
Since you framed the wall behind the in-concrete drain rough-in, you will use an S-trap from the bowl to the drain on the floor. Had you really wanted to use a P-trap, you would have framed the wall directly over the rough-in, used a P-trap into the wall, then an elbow down to the rough-in in the floor.
If you really want to use a P-trap then you'll need to move your wall. Since that does seem quite rather impractical (and probably makes the toilet rough-in unusably close to the wall), just stick with the S-trap. Functionally, an S- & P-trap work the same. I'm not aware of any additional clogging issues with an S-trap over a P (I have all S in my house and have very minimal clogging issues, especially since the long-haired daughter moved out on her own), so there's nothing to worry about there.
DroppingPlacing a vanity over the pipe in the floor really isn't that difficult in comparison to pushing it up against the wall. You have to measure carefully in each case to get the hole cut (in the floor or the back, as appropriate) in the proper place, but you just make the hole a bit oversized and it's not difficult in either case. There's no requirement that the pipe be perfectly centered for either installation method.