Can I have Protection and Fire Protection on the same piece?
No. The four protections form an exclusive group. A second book from that group costs 1 level and is not applied. To vary, use a different protection on each piece.
Which enchantments exist only on certain pieces?
Respiration and Aqua Affinity only on a helmet (and turtle shell). Feather Falling, Depth Strider, Frost Walker and Soul Speed only on boots. Swift Sneak only on leggings.
Is Thorns worth it?
That is opinion and the tool does not measure it. The price is fact, though: Thorns III costs 12 levels as a book on a fresh piece, three times the price of Protection IV (4).
Why is the tree plan cheaper?
Because the piece accumulates prior work penalty with every use (0, 1, 3, 7, 15, 31). Combining books with each other first keeps the piece at fewer uses when it receives the expensive books.