I suspect your brother is being infantilized, and his "bad behavior" is (at least in some cases) a normal reaction to how he is being treated.
You referred to a 20 year old as a "kid". This may be a translation error (e.g. from "kid brother" meaning younger brother), but it might be revealing that your brother is being infantilized by his family. I want to strongly point out here that I cannot know this for certain, this is just a possible inference.
If that inference is correct, this can lie at the root of this issue. If you get treated like a child, you're liable to start acting like a child.
To showcase the point I'm trying to make, I'm going to play devil's advocate here. The below points I touch on are mostly focused on explaining how your family's behavior towards your brother can steer his behavior. That doesn't put the responsibility solely on your family's shoulders, but it does show that this problem likely runs deeper than just your brother choosing to behave immaturely solely because "he wants to".
Lack of information
When I asked him to cut cucumbers for lunch he started cutting them without peeling them first and cut too large pieces
The first thing to notice here is that you didn't mention specifically asking for finely cut and peeled cucumber. I assume you didn't specify a size for cutting.
Your brother was left to decide the size of the cut himself, and the decision he made ended up being wrong.
The reason I mention this is because I never peel cucumbers when I cut them. Therefore, had you asked me, I would've know to ask you what size of cut you wanted, but I wouldn't have peeled the cucumber (or asked) either.
When dad asked about the large pieces
I'm going to take you at your words here, which is a bit of an assumption. If your dad asked why the cucumber was cut at that size, instead of pointing out that the cucumber should be cut at [this] size, this again makes it very hard for your brother to realize what it is you are expecting of him.
He said "Then do it yourself. Don't ask me to do it".
If you put someone in front of a challenge, and then don't give them the tools or information to tackle that challenge, then they're going to not tackle the challenge. Especially when people are inexperienced, they lack the self-confidence to persevere against unexpected complications.
Not telling your brother how the cucumber should be cut, is essentially the same as not giving him a knife to cut the cucumber with. While he could proactively ask about the size, he could proactively go out and find or buy a knife as well.
But that's not really the point here. You gave him a simple task, but made it harder for him than it needed to be. When your dad then indicated that he failed to meet expectations (i.e. peeling and finely cutting), if his feedback amounted to "this isn't good" rather than "I would've preferred having it [this way]", then you're partly to blame for causing your brother's failure to prepare the cucumber correctly.
Being overridden and ignored
we were invited to some wedding reception and dad asked him to come (his exams were next day), or mom has to cook dinner for him only. He denied and after some heated argument between him and dad he told if he goes to wedding he won't take his upcoming final exams.
Given that a heated argument ensued, I'm going to go ahead and guess that your dad wasn't taking "no" for an answer. That's not asking someone, that's commanding someone. Language is very important, and if you misrepresent what happened, you're going to misinterpret the things that happen because of it.
What is notably missing from this explanation is your brother's argument as to why he didn't want to go to the wedding. I suspect that he refused to go because of his exam. It makes sense given his eventual response of not taking the exam if he has to go to the wedding.
There's also plenty of room for compromise here, but you mention none of it. Did you suggest that your brother can stay home but needs to make/arrange his own dinner? Did you suggest that he comes to the wedding but will be provided the ability to retreat to a quiet place to study while there?
Assuming you weren't taking no for an answer, you're not representing your brother's case, and there's no mention of any attempted compromise; I feel like there's little attention being dedicated to your brother and his responsibilities, which is a core symptom of infantilization.
Children don't get to make important decisions, they get told what they have to do and adults tend not to engage in discussions when they've already made up their mind. This sounds exactly like what happened here with your brother. He was completely excluded from the decision making process. That's not how you treat an adult.
Your brother's reason for not wanting to go to the wedding is actually a very reasonable argument to make. He took responsibility for his exam by being able to prepare for it during that night, but his plans were overridden by your father's plans.
At the very core, your family communicated to your brother that it was more important that your mother doesn't cook than it is for him to properly prepare for his exams. To your brother, this comes across as your family putting your brother low on the list of priorities. Failing the exam has a much bigger consequence than the effort of preparing a single meal, yet the preparation of that single meal was treated as a bigger inconvenience.
So, from your brother's point of view, your family sets him up to fail his exam. Which then explains his response. If you're setting him up to fail (or he genuinely feels like you are), then he loses interest in actually taking the exam that you've set him up to fail at.
What especially strikes me in this example is that taking responsibility to study for an exam is mature behavior, and quite the opposite from the immaturity you accuse your brother of. As a teenager, I would've happily dropped my school work if I could blame it on having to go to a wedding, and so would anyone else in my class. The fact that he actually pushed back against this proves that he is taking charge of his own life and making sure that he achieves the task that is set before him.
Blocking him from studying when he indicates that he wants/needs to is in essence the same as asking him to cut a cucumber without telling him how to cut it. You're actually blocking your brother's ability to succeed at the task that is put before him. He is unable to both engage with his task and his family, because his family places obstacles between him and the task. Therefore, he feels forced to choose between his task or his family, and after repeated insistence from your family to engage in the obstacles to his task, he abandons the task itself.
In both the example of the cucumber and the exam, after repeated insistence from your family's side, your brother chooses to abandon the task (cucumber/exam) because of the insistent obstacle (not giving information/not being able to study due to going to a wedding) that is put before him.
My brother is unemployeed, he doesn't even try looking for job
You also mentioned he's studying in college for 3-4 hours and not yet counting any time for studying or tasks. From my western perspective, that is most of a full day's work. Given that he's living at home, it seems normal for him to not specifically have to go out looking for a job.
I can't judge Indian culture and I won't try, but I do wonder if "unemployed" is the right description for someone who is an active college student who, as established before, really works towards his education (given he makes sure to prepare for his exams).
This isn't right vs wrong
I'm not implying that your family is completely at fault or that your brother deserves no blame. But it does seem like this is a feedback loop, where your family treats your brother based on his behavior and your brother's behavior is influenced by how your family treats him.
There are other instances as well, like he never picks up call, doesn't answer when I ask him something or doesn't help with household chores (he goes to college for just 3-4 hours).
For the other behaviors you mentioned, there is not enough context to be able to attribute blame to anyone specifically.
In the interest of playing devil's advocate to highlight your brother's (potential) point of view, there are some relevant questions that come to mind, based on the details I've gathered from your previous (more detailed) examples:
- When your brother doesn't pick up the phone, is he busy doing something, or is he free to talk but willfully avoiding talking to you? How often do you call him, and if frequent, how important are these calls?
- When you ask a question, is it an actual question, or a command in disguise? Is your brother already able to answer that question after having heard it, or are you expecting him to take action in order to be able to find you an answer?
- Your brother goes to college for half the day, but how much time does he spend studying and working on tasks for college?
- Has your brother been asked to help with household chores in a manner that is constructive (explaining him how to do it) and not just imperative (commanding him, without further information)?
Based on the information you presented, I'm fairly convinced that you and your parents are somewhat oblivious to both your adult brother and how you treat/communicate with him. By infantilizing him, you inherently put him in the role of a child in your family structure, even though he is a 20 year old adult.
The subsequent question of why he's behaving childishly is a normal consequence of repeatedly treating him like a child.
What I suggest you to here is to communicate clearer. Assume your brother is of good will, but lacks practical experience.
- If you ask him to cut vegetables, tell him how they should be cut before he starts. Be precise. Don't expect him to infer what you may or may not be expecting. Eventually, he will learn to ask for information. But this takes time and experience.
- Rather than tell him where he will go with your family, ask him if he wants to come. When he responds, listen to what plans he already had, and try to understand that changing his plans impacts his life too. Try and find a compromise that works. For the example of the exam, you could've suggested that he fixes (or orders) his own dinner that night, or try to find him a quiet and secluded spot at the wedding so he can study in peace.
- Don't refer to him as a child or immature. Repeatedly being called something makes you believe that you are that which people call you. If you want him to become a mature adult, treat him like one.
- When he fails to do something correctly, rather than pointing out his failure, try to assist him and either teach or help him with making it better.
- Treat him like an adult. Respect that he has his own life schedule, the fact that he might not be available every time you call him, that he has certain obligations (college) that you can't just override at will, ...