When used as a noun, 'cocky' can mean 'a farmer whose farm is regarded as small or of little account'. Therefore, I suspect that the meaning of 'cocky' as an adjective stems from people's stereotype that a farmer of this kind is 'cocky' or arrogant.
Considering that, is the adjective 'cocky' discriminative? If a word/phrase can be used as both an adjective and a noun, is it often the case that people prejudice the noun to have the property of the adjective?
These words are not related. "Cocky" as an adjective comes from "cock" (ie a male chicken) that tend to act in a dominant and show-off manner.
"Cocky" as an noun is dialect, found only in Australian/NZ English, and is a shortening of "cockatoo farmer" (refers to the practice of working a small patch of land for a short period before moving on, in the manner of a feeding cockatoo) It has no connection in meaning or metaphor to the adjective.
Don't use "cocky" as a noun, not because its rude, but because unless you are in the Australian outback, nobody will understand you!
Feel free to use "cocky" in casual contexts, it is a mild insult, about as strong as "silly".
You have focused your attention on a usage of this noun that is specific to a small community. Most native speakers are not even familiar with the usage.
The usual meaning is this, which I have borrowed from the Cambridge Dictionary
Very confident, usually in a way that is slightly annoying:
Let's not get too cocky—things could still go wrong.
When I got my first job I was just a cocky teenager.
Wiktionary supplies this etymology:
From cock (“male domestic chicken”)
Many native speakers refer to a cock as a rooster. It is important to know that roosters are capable of breeding; thus, they have all their male hormones and exhibit elaborate courtship behaviors. The meaning of cocky derives from these behaviors.
There is no sense of discrimination in the word. It is simply a mild insult and can even be used humorously among friends and family.
If a word/phrase can be used as both an adjective and a noun, is it often the case that people prejudice the noun to have the property of the adjective?
I don't believe such a general rule exists.
Obviously in some cases the adjective and the noun are versions of the same word (such as "tough" - in its noun form it more or less means "someone who is tough"), and in these cases it makes sense that people consider both words to have similar properties.
But in other cases, with no semantic or etymological link between the two words, they're just different words that happen to be homonyms. Given that they occupy different spaces grammatically, such that only one of the meanings would make sense, it would be unlikely that people would consider the other word at all.
(And this assumes people even know of the other meanings. I'm a native BrE speaker and I've never heard of the use of "cocky" that you describe. It may be very niche or subculture-specific.)
