The Oxford English Dictionary (OED) acknowledges "the natural association of the colour green is with something positive," that is, with growth, and with "verdure, freshness, newness, health, and vitality", which association it says is "widespread among the Germanic languages", of which English is one. The word green is "borrowed from Germanic," says the OED. Of green in Germanic languages, the OED says that green is
Cognate with Old Frisian grēne , grēn green, fresh, Middle Dutch groene , groen , grone , grune green, fresh, youthful, inexperienced, (of food) raw, untreated, also groene , groen , grone , grune (noun) green, greenness, greenery, verdure, grassy ground, vegetables, greens, green fabric (Dutch groen ), Old Saxon grōni green (Middle Low German grȫne green, (as noun) green, greenness, greenery, verdure, grassy ground, fruit, vegetables, greens), Old High German gruoni , gruone , gruani , cruone green (as the colour of plants, some precious stones, the sea), fresh, healthy, (of food) raw, untreated, also gruonī (noun) green, greenness, greenery, verdure, health, vigour (Middle High German grüene , German grün ), Old Icelandic grœnn green (as the colour of verdure, etc.), fresh, hopeful, good, Old Swedish grön green, fresh, new (Swedish grön ), Old Danish grøn green, fresh (Danish grøn)
Note cognate does not mean "borrowed", it means "Descended from the same original language; of the same linguistic family. Of words: Coming naturally from the same root, or representing the same original word." (OED). In the case of green and grass and growth they all descend from Old Germanic root *gra- : grô- , which proto word *gra- is cognate with the Old Aryan *ghrā- to grow). So we're on solid grounds to say that there is an etymological association of green with grass and growth.
In addition, people take it as idiomatic that grass is green, even if "on the other side", where it is always said to be "greener" and in the axiom "as the grass is green" (example: "surely as the grass is green...")
The question, as asked, using words from the actual OP:
But, in English, (besides its use as growth) [green] is often symbolic of something negative. Why is this?
The OED acknowledges this "quite different and somewhat contrary development" and notes that it
probably result[s] in large part ultimately from association with ancient Greek χλωρός green, pale (see chloro- comb. form1) and with ancient and medieval medical traditions; compare classical Latin viridis ‘green’ used of a ‘greenish’ complexion taken as a sign of illness or excess of bile.
Speaking of bile, note that a reflection of the Latin viridis is contained in the English word biliverdin, "a green bile pigment that is formed by catabolism of hemoglobin and converted to bilirubin in the liver." The -verdin in biliverdin comes from viridis.
We have the disease cholera, which etymology online says is ultimately from this same khloros (that is, χλωρός) "pale green, greenish-yellow (my emphasis)." The related Greek term khole means bile. And the "The excretions in this disease [cholera] were freqently described, especially, in early use, as containing large quantities of bile" (OED).
As does jaundice, which since the Middle Ages was also called green jaundice
a disorder accompanied by greenish discoloration of the skin (cf. green sickness n. 1); specifically a form of jaundice in which there is accumulation of the green bile pigment biliverdin, usually as a result of biliary obstruction.
What about this green sickness? This was one of the most common diseases in the Middle Ages; it remained so until the 19th century when it was not diagnosed anymore and it has since become the forgotten disease. Green sickness was also called chlorosis, again pointing to the Greek influence.
Under green disease the OED says:
A disease characterized by greenish discoloration of the skin =chlorosis
and gives references from 1547.
Under chlorosis the OED says:
A disorder believed to occur almost exclusively in young, virginal women soon after puberty, characterized by a greenish pallor of the skin, cessation or irregularity of menstruation, and weakness, often accompanied by pica or other disturbance of appetite; an instance or case of this. Also called green sickness.
So yes, half the human race, that is, all young virgins could succomb to green disease, which was also equated with love sickness. It’s Not Easy Being Green: 3 Cures for Green Sickness, “The Virgin’s Disease” says that
Green sickness was so common, many women had cures written in their home recipe books.
The same site, dependent on Menstruation and the Female Body in Early Modern England relates that one cure to green sickness was copulation (with one's husband); that is: virgins needed to get married and consummate the marriage. (See also The Disease of Virgins: Green Sickness, Chlorosis, and the Problems of Puberty.)
Note that even Shakespeare's Juliet was admonished by her father ('Capulet')
To go with Paris to Saint Peter's Church
Or I will drag thee on a hurdle thither.
Out, you green-sickness carrion! out, you baggage!
(Act III, Scene 5)
Shakespeare also refers to the "green-sickness" in Antony and Cleopatra (III, 6), Pericles (IV, 6), and Henry IV Part 2 (IV, 3).
The juxtaposition of green with pale, and one supposes illness, occurs in MacBeth Act I, Scene 7:
Was the hope drunk
Wherein you dress'd yourself? hath it slept since?
And wakes it now, to look so green and pale...
And Charles Dickens writes a letter in 1843:
I am at this moment deaf in the ears, hoarse in the throat, red in the nose, green in the gills,..and fractious in the temper from a most intolerable and oppressive cold.
which is the first recorded example in the OED of green around (also about, at, in) the gills. At the time of the letter, the use of green to mean sickness was well established.
As far as other "negative things," such as jealousy, envy, they flow from the same "development" (see above) as do sickness and illness:
Of the complexion: having a pale, sickly, or bilious hue, indicative of fear, envy, ill humour, or sickness (also in green and wan, green and pale).
The use of green-eyed jealousy meant or implied being sick with it. If someone is 'green with envy' they are 'sick with envy'.