You would do well to incorporate an elbow or elbows, if possible into the part of the drain line that is behind the white PVC pipe, the metal pipe. The trouble with that would be is that old of pipe is easily damaged (broken) at the joints where the threaded sections are.

The reason I suggest this as the place to tie in the new extension is that by the time, I feel, if you simply replace the drain it will be so far away from the wall it will be a number of inches away from the wall it is against now. To remedy that, it would take so may elbows to get it back to the wall, I am afraid the drain would not drain readily or perhaps clog easily. So back at the reason why I mentioned taking it back and tying it into the farther in from the joint where it is tied in now. The place I refer to is at the wye, there is a small straight pipe there, perhaps there is a smooth part (no threads) to it to use a neoprene coupler there.

No, you should not do that. As mentioned in a comment, the drain is going to fill up with gunk and clog. Depending on use, you'll find that you have to snake this line quite often.

While traps do hold water, they are limited in the depth of water they can hold. Traps are also designed in such a way, that normal use stirs up any "sediment" and helps flush it away (self cleaning).

Uniform Plumbing Code

Chapter 10 Traps and Intercepters

1003.0 Traps - Described

1003.1 General Requirements. Each trap, except for traps within an interceptor or similar device shall be self-cleaning. Traps for bathtubs, showers, lavatories, sinks, laundry tubs, floor drains, urinals, drinking fountains, dental units, and similar fixtures shall be of standard design, weight and shall be of ABS, cast brass, cast iron, lead, PP, PVC, or other approved material.

I'm not sure how far away the vent is for this fixture. If that's 2" pipe, it has to be within 8' of the trap weir. If it's 1½" pipe, the vent has to be within 6' of the trap weir.