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This cactus is very slow growing, but can reach a 10 foot diameter in 150 years. If it has been planted too close to it's neighbor, someone didn't take later growth into consideration.
To trim an arm, (branch) on this cactus, you should use a very sharp saw and cut as close to the base of the plant as possible.
If you are asking about trimming the entire plant, this is what I would suggest. If the plant is too tall, and there is a subspecies that grows to only 10' tall, take out the tallest arms. DO NOT saw off the arms to shorten the plant. The plant will initiate new growth at the injury site. The new growth will be too heavy and weigh the arm down, causing more injury, or an unsightly appearance.

This is likely Scale pests.

They can be scraped away from the plant.

Neem Oil can also be used to treat Scale.
Here are some links with more information.
https://www.gardeningknowhow.com/ornamental/cacti-succulents/organ-pipe-cactus/organ-pipe-cactus-care.htm
https://www.gardeningknowhow.com/plant-problems/pests/pesticides/neem-oil-uses.htm

The Organ Pipe Cactus does turn grey as it matures and ages.

This article has more information.

https://www.gardeningknowhow.com/ornamental/cacti-succulents/organ-pipe-cactus/organ-pipe-cactus-care.htm

Yes, you can propagate your Cactus piece.

This article will help you with this task.

https://extension.arizona.edu/sites/extension.arizona.edu/files/pubs/az1483.pdf

This actually sounds like a case of over-watering. These are very drought tolerant, and will not tolerate "wet feet" for very long. I would let the potting material dry out pretty well in between waterings. You might apply a few drops of rosemary oil in the water for the next watering, just in case an infection is trying to start in.

Here is an article for more information on this cactus: https://www.gardeningknowhow.com/ornamental/cacti-succulents/organ-pipe-cactus/organ-pipe-cactus-care.htm

It looks like it has sustained damage at one time. If this spot doesn't spread, then it will not harm the cactus. If it is spreading, then it could be an infection. If this is the case, then I would recommend applying peroxide to the wound. Also use some wettable sulfur around the base according to the instructions on the package.