Old English hebban "to lift, raise; lift up, exalt" (class VI strong verb; past tense hof, past participle hafen), from Proto-Germanic *hafjan (source also of Old Norse hefja, Dutch heffen, German heben, Gothic hafjan "to lift, raise"), from PIE *kap-yo-, from root *kap- "to grasp." The sense evolution would be "to take, take hold of," thence "lift."
Related to have (Old English habban "to hold, possess"). Meaning "to throw" is from 1590s. Nautical meaning "haul or pull" in any direction is from 1620s. Intransitive use from early 14c. as "be raised or forced up;" 1610s as "rise and fall with alternate motion." Sense of "retch, make an effort to vomit" is first attested c. 1600. Related: Heaved; heaving. Nautical heave-ho was a chant in lifting (c. 1300, hevelow).
Old English forst, frost "frost, a freezing, frozen precipitation, extreme cold," from Proto-Germanic *frustaz "frost" (source also of Old Saxon, Old Frisian, Old High German frost, Middle Dutch and Dutch vorst), from the verb *freusanan "to freeze" (source of Old English freosan "to freeze"), from suffixed form of PIE *preus- "to freeze; burn" (see freeze (v.)). Both forms of the word were common in English till late 15c.; the triumph of frost may be due to its similarity to the forms in other Germanic languages. A black frost (late 14c.) is one which kills plants (turns them black) but is not accompanied by visible frozen dew.
mid-15c., "weight, heaviness, quality of weight," from heave (v.) on analogy of thieve/theft, weave/weft, etc. Also influenced by heft, obsolete past participle of heave.
1640s, of hair, "turning white;" 1680s, of glass, "having a rough and unpolished surface;" 1734 in cookery, "covered with something (sugar, icing) resembling frost," past-participle adjective from frost (v.).