New hatchet quests

Similar quest reqs to those of the pickaxe of life and death, but Pieces of Hate instead of Cabin Fever.

Yes and yes.

Quest reqs of https://runescape.wiki/w/Pickaxe_of_Life_and_Death#Creation but replace Cabin Fever with Pieces of Hate.

The hatchet/pickaxe of earth and song is the part that needs Plague's End (level 70 crystal hatchet/pickaxe and completion of the elf questline for "song"), Birthright of the Dwarves (for level 80 imcando hatchet/pickaxe and completion of the dwarf questline for "earth") and While Guthix Sleeps for a tertiary ingredient that combines them into the level 90 hatchet/pickaxe of earth and song.

The level 100 hatchet/pickaxe of life and death is then an additional upgrade of the hatchet/pickaxe of earth and song with extra requirements (such as pirate quests and necromantic elements).

If you don't want to do quests and are happy to settle for not having an augmentable hatchet, though, you'll be able to smith hatchets up to level 100 primal, in a similar manner that you can smith pickaxes. (We're adding smithable hatchets from orikalkum upwards.)