Old English hof "hoof," from Proto-Germanic *hōfaz (source also of Old Saxon, Old Frisian hof, Old Norse hofr, Danish hov, Dutch hoef, German Huf "hof"), perhaps from PIE *kop- "to beat, strike" (source also of Sanskrit saphah "hoof," Polish kopyto "hoof;" see hatchet (n.)). But Boutkan acknowledges only Indo-Iranian cognates and writes, "We may be dealing with a typical relic form that only survived in the periphery of the IE area ...." For spelling, see hood (n.1).
A hoof differs from a nail or claw only in being blunt and large enough to inclose the end of the limb; and almost every gradation is to be found between such structures as the human nails, or the claws of a cat, and the hoofs of a horse or an ox. The substance is the same in any case, and the same as horn, being modified and greatly thickened cuticle or epidermis. [Century Dictionary]
Hoof-and-mouth disease is attested from 1866. Phrase on the hoof is from 1750 as "walking;" later it was cattlemen and butchers' slang for "not (yet) slaughtered."
beat n.
c. 1300, "a beating, whipping; the beating of a drum," from beat (v.). As "throb of the heart" from 1755. The meaning "regular route travelled by someone" is attested from 1731, also "a track made by animals" (1736), from the sense of the "beat" of the feet on the ground (late Old English), or perhaps that in beat the bushes to flush game (c. 1400), or beat the bounds (1560s). It was extended to journalism by 1875.
The musical sense is by 1842, perhaps from the hand motion of the conductor and the notion of "beating the time":
It is usual, in beating the time of a piece of music, to mark or signalize the commencement of every measure by a downward movement or beat of the hand, or of any other article that may be used for the purpose .... ["Godfrey Weber's General Music Teacher," 1842]
Earlier in music it meant a sort of grace note:
BEAT, in music, a transient grace note, struck immediately before the note it is intended to ornament. The beat always lies half a note beneath its principal, and should be heard so closely upon it, that they may almost seem to be struck together. ["The British Encyclopedia," London, 1809]