bill (v.2) Look up bill at Dictionary.com
"to stroke beaks," as doves do, hence, of lovers, "caress fondly," 1590s, from bill (n.2)). Paired with coo since at least 1764; Century Dictionary [1902] defines bill and coo (by 1768) as "to kiss and caress and talk nonsense, as lovers." Related: Billed; billing.
bill (n.1) Look up bill at Dictionary.com
"written statement," late 14c., "formal document; formal plea or charge (in a court of law); personal letter," from Anglo-French bille, Anglo-Latin billa "a writing, a list, a seal," from Medieval Latin bulla "decree, seal, sealed document," in classical Latin "bubble, boss, stud, amulet for the neck" (hence "seal"); see bull (n.2).

Sense of "written statement detailing articles sold or services rendered by one person to another" is from c. 1400; that of "order addressed to one person to pay another" is from 1570s. Meaning "paper intended to give public notice of something, exhibited in a public place" is from late 15c. Sense of "paper money, bank-note" is from 1660s. Meaning "draft of a proposed statute presented to a legislature" is from 1510s.
bill (n.3) Look up bill at Dictionary.com
ancient weapon, Old English bill "sword (especially one with a hooked blade), chopping tool," from Proto-Germanic *bili-, a word for cutting or chopping weapons (compare Old Saxon bil "sword," Middle Dutch bile, Dutch bijl, Old High German bihal, German Beil, Old Norse bilda "hatchet"), possibly from PIE root *bheie- "to cut, to strike" (source also of Armenian bir "cudgel," Greek phitos "block of wood," Old Church Slavonic biti "to strike," Old Irish biail "ax").
bill (n.2) Look up bill at Dictionary.com
"bird's beak," Old English bill "bill, bird's beak," related to bill, a poetic word for a kind of sword (especially one with a hooked blade), from Proto-Germanic *bili-, a word for cutting or chopping weapons (see bill (n.3)). Used also in Middle English of beak-like projections of land (such as Portland Bill).
bill (v.1) Look up bill at Dictionary.com
"to send someone a bill of charge," 1864, from bill (n.1). Related: Billed; billing.