I do peel my carrots as I can't get them as clean as I like (unless I am using baby carrots, which are actually big carrots peeled, cut and shaped to look like baby carrots).
When you say "mash" do you mean actual mashed potato with butter, milk, etc.? I am not sure how this would thaw after frozen- I don't think it would end up fluffy but should still taste fine. I froze pureed potato veggie cubes for dd and they turned out fine but puree is so different from mashed. I also froze baked potatoes with cheese where you scoop out the potato and mix it with seasoning and cheese and then fill it, and these work out fine when defrosted.