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I'm not exactly sure what you mean by the "mass close to the trunk." If you're referring to the uncut pieces of frond that were left behind when you pruned a frond, yes, you should go ahead and scrape or cut these down flush with the trunk. You would save yourself this work, and create a more lovely tree, if in the future when you prune a frond, you cut it flush with the trunk to begin with. Here is an article with some more information on Phoenix roebelini that might be of use to you: https://www.gardeningknowhow.com/houseplants/pygmy-palm/grow-pygmy-palm-trees.htm

I can't give you very much information because I don't know what kind of palm you have, what kind of environment it's in, size, age, etc. If you know what kind of palm it is, research that; if you don't, try to find out, or send us a picture so we can give you some better info. Here are a few basics for palm care: https://www.gardeningknowhow.com/houseplants/palms/growing-palms-indoors.htm

Yes it certainly can. It's no big job for the lawncare people to put mulch around the base of the palms so that lawn trimming can be done without hitting the palm trunks, and you're within your rights to request that this be done.

The tree is toast. Palms grow only from a central growing point located at the top of the trunk, in the center of the rosette of leaves.

I am sorry to say that I am not familiar with that combination. Liquid copper is an effective fungicide, so it is possible that he recognized that what was wrong with your trees was fungus based and gave you a mixture of fungicides. Soap is a common ingredient in mixtures because it helps it stick better. Baking soda is another common fungicide. Do you remember if the other items were something you would have bought at the nursery or were they household items?

This article should help you: https://www.gardeningknowhow.com/ornamental/trees/palms-trees/propagate-palm-pups.htm

In the spring, gently cut the shoots away from the mother plant, making sure to take some root with it. Sometimes it helps to dig out around the pup/shoot while you are doing this so you get a better idea of how and where everything is. Then, you can pot them up and keep them well watered. Once you see it putting out growth, it can be transplanted into the ground.