When you have to answer psycho questions like that, it is not good. Ideally you want to get to the point in your expertise level where BS like that is not a factor. Unfortunately, when you are just out of school, there is not a lot to differentiate you from a million other similar people, so companies start making decisions based on nonsense trick questions like these, so let's take them one at a time:
Tell about a time you have been in conflict with a coworker.
I have had good relations with my coworkers.
(Surely you have had some conflict? No, I haven't yet, thank goodness.)
There is no reason at all to even answer this dumb question in any way other than what I written above, especially when you are young and have little job experience.
Tell about a time you disagreed with your supervisor.
I have only had one/two supervisors so far, and have not had occasion to disagree with them. In general, both were very reasonable people and the job I was doing was very straightforward, so I did not have any problems.
Basically the same as the previous question.
What was the biggest failure in your last job position?
There were no failures. We were writing software to add printing to the report generator [or whatever] and finished on time, a little early actually. The boss was very happy with it.
Once again, no reason to even go there.
What are you looking for in your next job?
I am hoping to work with a good team to write software modules that will help the company be successful with their new products.
Just tie your personal goals to the company goals so they match. Your goal is help the company meet their goals.
I would like to give you some general advice about how to deal with the mind game type questions you are asking about, the type that fits into the pattern of "Tell me why you suck". As I have written above, you basically deny it in all instances. "I do not suck." I have given examples of how to express this in a polite way above.
Let's consider what is behind such questions in greater depth. When high level CIA or KGB agents are being trained, they are taught how to resist interrogation by the enemy, should they be captured. This training is very complex and arduous, but the first thing they are taught, the cardinal rule in all instances is: never confess. The HR rep or whoever is asking questions like those above is essentially acting like a penny ante interrogator; they are soliciting a confession from you. Dealing with such interrogation is simple: never ever confess. It's a simple rule, obey it always.
Just to make this crystal clear for you... Let's imagine you have some nuanced 30-minute conversation about how you handled a conflict with a coworker. Do you really think the HR rep is going to write "Candidate seems to be masterful and diplomatic at dealing with difficult coworkers." No, of course not; the rep is going to write, "Candidate told me he got into a fight with a coworker at his last job." You just got voted off the island. All the delicate subtleties you are describing to the HR rep will get distilled down to one basic fact: you had a fight. When an HR rep has 10 candidates that all have identical credentials for 1 position, what do you think they are going to do? Look for reasons to eliminate people.