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."
mark n.1
"trace, impression," Old English mearc (West Saxon), merc (Mercian) "boundary, limit; sign, landmark," from Proto-Germanic *markō (source also of Old Norse merki "boundary, sign," mörk "forest," which often marked a frontier; Old Frisian merke, Gothic marka "boundary, frontier," Dutch merk "mark, brand," German Mark "boundary, boundary land"), from PIE root *merg- "boundary, border." Influenced by, and partly from, Scandinavian cognates. The Germanic word was borrowed widely and early in Romanic (compare marque; march (n.2), marquis).
The primary sense "boundary" had evolved by Old English through "pillar, post, etc. as a sign of a boundary," through "a sign in general," then to "impression or trace forming a sign." Meaning "any visible trace or impression" is recorded by c. 1200. Meaning "a cross or other character made by an illiterate person as a signature" is from late Old English. Sense of "line drawn to indicate the starting point of a race" (as in on your marks..., which is by 1890) is attested by 1887.
The Middle English sense of "target" (c. 1200) is the notion in marksman and slang sense "victim of a swindle" (1883). The notion of "sign, token" is behind the meaning "a characteristic property, a distinctive feature" (1520s), also that of "numerical award given by a teacher" (by 1829). To make (one's) mark "attain distinction" is by 1847.
In medieval England and in Germany, "a tract of land held in common by a community," hence Mark of Brandenburg, etc.