Kitchens – Should I add a grease trap to a long drain run from the kitchen

Much too much up and down! You want a P trap with an additional angled connector toward the wall.

The first down and up from the sink drain is correct. That is the P trap. It should end with a horizontal leg pointing toward the wall (an L shaped piece).

This will not line up with the opening in the wall because the drain is offset. Therefor that leg will run a distance and then will use an angle to bend it back toward the wall.

You could use a right angled slip joint connector (running horizontally).

A 45 degree angle would be better (the less sharp the bend, the less likely to stop up). However it may be hard to fine a 45 degree slip joint elbow. You could use standard pipes and elbows, but these would require glued joints rather than slip joints.

Finally there should be a straight run from the angled connector to the wall connection.

You can rotate the P part of the trap so the L leg coming off the P is parallel to the wall (assuming you are using the 90 degree elbow; if not, adjust accordingly). Then the elbow turns the drain toward the wall. Another horizontal run carries the drain back to the wall.

You need to measure the length of the horizontal leg of the P and the length of the connecting run to the wall; be sure to allow for the sections embedded in the elbow and trim each accordingly. Assemble.

It is easier done than described.

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.