Old English habban "to own, possess; be subject to, experience," from Proto-Germanic *habejanan (source also of Old Norse hafa, Old Saxon hebbjan, Old Frisian habba, German haben, Gothic haban "to have"), from PIE root *kap- "to grasp." Not related to Latin habere, despite similarity in form and sense; the Latin cognate is capere "seize.
Sense of "possess, have at one's disposal" (I have a book) is a shift from older languages, where the thing possessed was made the subject and the possessor took the dative case (as in Latin est mihi liber "I have a book," literally "there is to me a book"). Used as an auxiliary in Old English, too (especially to form present perfect tense); the word has taken on more functions over time; Modern English he had better would have been Old English him (dative) wære betere.
To have to for "must" (1570s) is from sense of "possess as a duty or thing to be done" (Old English). Phrase have a nice day as a salutation after a commercial transaction attested by 1970, American English. Phrase have (noun), will (verb) is from 1954, originally from comedian Bob Hope, in the form Have tux, will travel; Hope described this as typical of vaudevillians' ads in Variety, indicating a willingness and readiness to perform anywhere.
'There are but two families in the world, as my grandmother used to say; "the Have's and the Have-not's," and she stuck to the former; and now-a-days, master Don Quixote, people are more inclined to feel the pulse of Have than of Know.' ["Don Quixote de la Mancha," transl. Charles Jarvis, London, 1742]
late Old English hæfen "haven, port," from Old Norse höfn "haven, harbor" or directly from Proto-Germanic *hafno- (source also of Danish havn, Middle Low German havene, German Hafen), perhaps from PIE root *kap- "to grasp" (source of have) on notion of place that "holds" ships. But it might rather be related to Old Norse haf, Old English hæf "sea" (see haff). Figurative sense of "refuge," now practically the only sense, is c. 1200. Havener "harbor master" is attested from mid-14c.
past tense and past participle of have, from Old English gehæfd. Assimilation of -f- to a following consonant is typical (as also in woman, lord, lady, head (n.), leman). Used since late Old English as an auxiliary to make pluperfect tense-phrases. You never had it so good (1946) was said to be the stock answer to any complaints about U.S. Army life.