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I see from the photo, that the soil is too moist. This has likely caused an infection, and it may be beyond repair at this point.

Cactus care requires a light mix made for cacti, that is fast draining. They like to be thoroughly dry between waterings in the future.

This collection of articles will give you more information on cactus care: https://www.gardeningknowhow.com/ornamental/cacti-succulents/scgen/

Uh-oh... This does appear to be a fungal infection. You will want to apply wettable sulfur the next time you water, and cut off any infected parts. Rub the spot with peroxide to make sure it kills the infection from the cut.

Make sure to only water when the soil is almost completely dry. You may even want to use a cactus mix type soil, if you aren't already.

Yes. You can remove the affected growth. I'm inclined to think that there could be a fungal or bacterial infection. Could you include a photo? This could help me to see what the problem is.

After you cut the top, make sure to wipe the cut with peroxide to kill of any infection. You should always makes sure to let the soil mix dry out almost completely between waterings. On the next watering mix a few caps full of peroxide into the water before using it.

Because many Cactus and Succulents have small root structures, growers often use this method to ensure the containers make it to the store shelves.
This is not a good growing and planting method for these plants. As you have discovered it can be difficult to remove the plant from the container and even difficult to water the plants.

This glue often can be softened or removed by wetting with water. You can remove it with a small spoon once it softens. Avoid over wetting the soil and the roots.
Once you have removed it you can repot with a cactus soil mix.

They do start out thin, but they should thicken up over time. Make sure that the soil is a light orchid or cactus mix, so that the roots can have plenty of air pockets. This article will give you more information: https://www.gardeningknowhow.com/ornamental/cacti-succulents/fishbone-cactus/orchid-cactus-care.htm

Many cacti look so much alike, that it can be nearly impossible to tell them apart. My best guess would be a species of trichocereus. I could suggest taking a sample, or a picture to your local extension service, and they will be able to help you further. Here is a link to find the nearest one to you: https://www.gardeningknowhow.com/extension-search/