If you examine the tailpiece on the left sink you will notice that it tilted about 15deg to the right. That angle will most assuredly result in a leak. The tailpiece must be straight down from the sink.

To get as bit more vertical clearance at the P-trap disconnect the cross tube from the left tailpiece and the trap. Cut the tailpiece off about 1-1/2" below the dishwasher inlet. Now re-connect the cross piece and it will be about 2" to 3" higher.

Now attach a flange type extension tailpiece to the drain on the right sink. (Be sure to install the clear plastic ferrule on the flange before you connect it to the sink).

You will have to cut both the cross tube and the tailpiece on the right sink to fit the baffled tee directly below the sink drain.

You will have to cut the black metal pipe. That pipe is 1-1/2" cast iron and it can be very difficult to cut if not done correctly. It can be cut with a hacksaw or reciprocating saw, but keep in mind that cast iron pipe is brittle and if you try cutting it with a saw you run a high risk of shattering the pipe, leaving a very sharp and rough end, and if your luck is like mine it would probable shatter back into the wall, not to mention that you will probably wear out half a dozen good quality blades before you get it cut.

The correct way to cut that pipe is with a cast iron pipe "snap cutter". You can rent a snap cutter from a local tool rental for about $15 a day, and I can assure you, if you try cutting with a hacksaw you will go through at least that much money in blades, not to mention an hour or more laying on your back in cramped quarters and cussing the whole mess.

The snap cutter has a cast iron body with a crank on one end and what looks like a bicycle chain on the other end. The chain is wrapped around the pipe at the point where you want to make the cut and re-attached to the cutter. You then turn the crank a few times until you hear a snap or pop sound and the cut is complete.

Once you have teh cast iron cut, discard both the cast iron pipe and the existing P-trap the use a Fernco coupling to connect a short piece of 1-1/2" sched 40 PVC pipe. Attach a PVC 1/16th bend(DWV grade 22.5deg elbow) on the pipe and another short length of PVC with a Trap adapter on the end of the pipe. Install a new 1-1/2" tubular slip type p-trap from the tailpiece under the baffled tee to the line to the wall.

"black metal pipe"

Not a great picture, but the exterior looks awfully smooth, like ABS.