early 13c., hachen, "to produce young from eggs by incubation," probably from an unrecorded Old English *hæccan, of unknown origin, related to Middle High German, German hecken "to mate" (used of birds). Meaning "to come forth from an egg," also "cause to come forth from an egg" are late 14c. Figurative use (of plots, etc.) is from early 14c. Related: Hatched; hatching.
"opening, grated gate, half-door," Old English hæc (genitive hæcce) "fence, grating, gate," from Proto-Germanic *hak- (source also of Middle High German heck, Dutch hek "fence, gate"), a word of uncertain origin. This apparently is the source of many of the Hatcher surnames; "one who lives near a gate." Sense of "opening in a ship's deck" is first recorded mid-13c. Drinking phrase down the hatch attested by 1931 (the image is nautical).
"engrave, draw fine parallel lines," late 14c., from Old French hachier "chop up, hack" (14c.), from hache "ax" (see hatchet). Related: Hatched; hatching. The noun meaning "an engraved line or stroke" is from 1650s.