Old English hat "hot, flaming, opposite of cold," used of the sun or air, of fire, of objects made hot; also "fervent, fierce, intense, excited," from Proto-Germanic *haita- (source also of Old Saxon and Old Frisian het, Old Norse heitr, Middle Dutch and Dutch heet, German heiß "hot," Gothic heito "heat of a fever"), of uncertain origin, perhaps related to Lithuanian kaisti "to grow hot;" both could be from a substratum word.
With a long vowel in Middle English (rhyming with boat, wrote) which shortened in modern English, perhaps from influence of comparative hotter. As an adverb, Old English hote.
Hot as "full of sexual desire, lustful" is from c. 1500; the sense of "inciting desire" is 18c. Taste sense of "pungent, acrid, biting" is from 1540s. Sense of "exciting, remarkable, very good" is 1895; that of "stolen" is first recorded 1925 (originally with overtones of "easily identified and difficult to dispose of"); that of "radioactive" is from 1942. Of jazz music or combos from 1924.
Hot flashes in the menopausal sense attested from 1887. Hot stuff for anything good or excellent is by 1889, American English. Hot seat is from 1933. Hot potato in figurative sense is from 1846 (from being baked in the fire coals and pulled out hot). Hot cake is from 1680s; to sell like hot cakes is from 1839.
The hot and cold in hide-and-seek or guessing games (19c.) are from hunting (1640s), with notion of tracking a scent. Hot and bothered is by 1921. Hot under the collar in the figurative sense is from 1895.