The funeral display is so called because it resembled a harrow (hearse in its sense of "portcullis" is not attested in English before 15c.). Sense extended to other temporary frameworks built over dead people, then to "vehicle for carrying a dead person to the grave," a sense first recorded 1640s. For spelling, see head (n.).
c. 1300, rehersen, "to give an account of, report, tell, narrate (a story); speak or write words;" early 14c., "repeat, reiterate;" from Anglo-French rehearser, Old French rehercier (12c.) "to go over again, repeat," literally "to rake over, turn over" (soil, ground), from re- "again" (see re-) + hercier "to drag, trail (on the ground), be dragged along the ground; rake, harrow (land); rip, tear, wound; repeat, rehearse;" from herse "a harrow" (see hearse (n.)).
The meaning "to say over again, repeat what has already been said or written" is from mid-14c. in English; the sense of "practice (a play, part, etc.) in private to prepare for a public performance" is from 1570s (transitive and intransitive). Related: Rehearsed; rehearsing.
"heavy one-handed metal weapon, often with a spiked head, for striking," c. 1300, from Old French mace "a club, scepter" (Modern French massue), from Vulgar Latin *mattea (source also of Italian mazza, Spanish maza "mace"), from Latin mateola (in Late Latin also matteola) "a kind of mallet." The Latin word perhaps is cognate with Sanskrit matyam "harrow, club, roller," Old Church Slavonic motyka, Russian motyga "hoe," Old High German medela "plow" [de Vaan, Klein].
As a ceremonial symbol of authority or office, a scepter or staff having somewhat the form of a mace of war, it is attested from mid-14c. Related: Mace-bearer.