What is the etymology of "嗨"? Is it used because it sounds like the English word "Hi"?
I wouldn't call 嗨 onomatopoeic, since it doesn't seem to be imitating any particular sound.
According to the Ministry of Education's 國語辭典, 嗨 is an interjection that expresses discouragement, regret, surprise, etc. It is sometimes written as 咳, (in this sense also read hai, not ke).
This use goes back to Yuan dynasty drama. It can also be used in chanting sort of expressions such as 嗨唷, which is what everyone was yelling when I was on the tug-of-war team. Acc. to 國語辭典, this use can also be found in Yuan dynasty plays.
Unlike some dictionaries, 國語辭典 does specifically mention the use of 嗨 as 音譯 for English Hi. This use must be quite modern. Acc. to the Oxford English Dictionary, the use of 'hi' as 'a word of greeting' (=hello) is primarily North American in origin, apparently late 19th cent. I wouldn't be surprised if 嗨 = Hi, was a post WW2 innovation in Chinese.
I tried looking for sources trying to find where 嗨 was to represent an English word.
近来，常常听到主持人用“嗨”这个字。。。令观众和听众只能朦胧地意会 (Recently, I've heard a lot of TV hosts use the word 嗨...it causes the listeners to only be able to guess at what the host means).
Because the article notes that much of the audience is probably unfamiliar with the word, the origin of 嗨 to mean any English word must have been before 2003, but not that much before.
This 2004 article says
Hi（嗨），苏珊 (Hi, Susan)
which helps support my previous conjecture about the timing of the meaning of 嗨 to mean
I think the term more commonly use to say
hi in Chinese is
嘿, but some time in the 90s or so, the word 嗨 started being utilized due to how similar it sounds to the English word
Also on another note, in addition to what wpt said, 嗨哟 is used like heave-ho and according to Wikipedia it was used in some old songs that provided the basis for 东方红.
口 + 屮 + 水 + 母 = mouth + sprouting plant + 3 points water + woman
Edit： Forgot the 三点水
http://www.zdic.net/z/16/zy/55E8.htm has a nice old version, you can see the woman clearly. Quite what the significance of her having a plant for a head is escapes me， （maybe someone here knows) and the representation of her bosom leaves a lot to be desired!
Here are some old versions of 海
zdic has as number 1 meaning 象声词, so I presume that is its main use.