I am also new to Affinity Photo. While I'm not experiencing the same problem as your first question, I would also like to know the answer to your second question and experience the same thing as your third question, to some extent.
For the brush categories, I would really like to be able to duplicate and move brushes around. There are so many good brushes bundled with Affinity that I'd like to be able to have a custom category with the brushes I like most so I won't have to search through the categories to find the specific one I'm looking for. Duplicating brushes would also allow me to make edits to those brushes without harming the original.
As for the brush quirk, I seem to experience this most when the brush has 100% pressure sensitivity to the size and the brush is huge. While it still happens when the brush size is small, it's much less noticeable. I'd also want to note that I'm using the iPad Pro with an Apple Pencil.