tl;dr character analysis funtimes
On Trust:
The Chief is not a very trusting person, but is at the same time somewhat gullible. He's used to assuming that the people in direct command of him may not have his or his unit's best interests at heart until proven otherwise. It's not always the case, but going in with that mindset has helped him deal with setbacks in the past when it has been and it's heavily ingrained into him to be initially a little suspicious of officers. This is not due to any particular malice on either side, but just practical experience. He's the same way with people in general, and he'll be wary toward them until he feels he has the measure of them. This can take anywhere from a few minutes to a few years, depending on circumstances and the person in question. This is only basic trust, though. He very, very rarely "opens up" to people and generally dislikes when someone tries to get close on a personal level. If the Chief trusts you he's fairly sure you will not do anything (out of will, ignorance, or incompetence) to get him killed, but he will most likely not tell you about his life or confide his problems in you.
His gullibility comes in that while he may not trust people initially, once he's evaluated them as "safe" his guard tends to go down quite a bit and it's easy to string him along. The prime example of this is 343 Guilty Spark. Spark helped the Chief fight the Flood and was not outwardly hostile toward him (at least initially), so the Chief very nearly fired the first Halo because he had decided it was safe to follow Spark's instructions. He was alone at the time, so he also had no one else's suspicion to influence his evaluation. If it hadn't been for Cortana bursting in and stopping him, he would have done it. As long as what you're telling him seems to make sense based on the information he has (all the better if it's information you've given him and time is short), the Chief is likely to believe you.
John and Kids:
The Chief's a little weird with kids, mostly because his own childhood was ripped violently away from him when he was a boy. They make him a touch uncomfortable because he's not sure what to do with them or what's expected of him. He's better with older children than with younger, and tends to treat them according to how they act. If you behave like an adult, the Chief will (mostly) treat you like one in turn. He will soften things for those he sees as more fragile, and be blunt and straight with the ones who've proven their maturity to him.
Perhaps surprisingly to those who do not know his background (read: almost everybody) the Chief has a good deal of confidence in the capabilities of kids with training. He himself was able to kill a man by eight and fighting in full on warfare against the Covenant at fourteen, so his view of this is a bit skewed.
He has a tendency to look out for younger people, though he's subtle about it. This is especially true in the case of those who have been flung into strange circumstances like his own. He sees something of himself in them. If called upon, John could likely easily take up the mantle as a mentor as Chief Mendez was to him. He is a firm but fair teacher. He will work you to the bone, but so help him you will learn something from it. And if it's something that keeps you alive, then he has done what he set out to do by it.
John and Relationships:
John is not close to people. This is a fact of life. Fire burns, water is wet, and the Master Chief is as much of an island as he can be. There are multiple reasons for this, one of the simplest and most obvious being that he has lost many people in his life and it's easier to get past if he doesn't have a strong bond to them. Another easy one is that his origins are so very, very classified. He takes no pleasure in lying, so it's just easier to avoid the questions altogether by keeping people at arm's length.
John's leadership position also isolates him, as he sees it as his responsibility to maintain his composure at all times for the benefit of everyone else. If people know him well, it's a possibility they'll be able to spot his own distress and it's not good for anyone's peace of mind. John's problems are in his mind his and his alone, and he will not readily share what he sees as his burden. He will tear himself apart inside, if it comes to it, to protect those in his charge.
This tendency toward self sacrifice is part of another reason why the Chief does not form close ties. It's not actively seeking it so much as that he sees himself as an acceptable loss in the line of duty. He has a job to do and he will pour absolutely everything he has into it. The Chief's life isn't his to promise to anybody, and he hates making promises he knows he can't keep. He belongs to the UNSC, he belongs to his superiors, and he belongs to his Spartans. He has a lot of responsibilities and there's little room for anything else. This especially factors into his unwillingness to have any romantic connections. He feels he'd be unable to provide for whoever it may be, be it in security, time, emotional openness, or shelter from the mess of his life they'd be inevitably pulled into. It's hard for him to even understand why someone would want to be with him, if they truly knew him. Couple that with the fact that he has so many things to hide already and so many barriers and how his continued life is never guaranteed, and it strikes him as a shamefully selfish action doomed to failure from the start. That way lies only heartbreak, and John knows it. And it's one of the few things that actually scares him.
Better he be alone on that inevitable day he's killed, and not pull anyone down with him.
The Chief is not a very trusting person, but is at the same time somewhat gullible. He's used to assuming that the people in direct command of him may not have his or his unit's best interests at heart until proven otherwise. It's not always the case, but going in with that mindset has helped him deal with setbacks in the past when it has been and it's heavily ingrained into him to be initially a little suspicious of officers. This is not due to any particular malice on either side, but just practical experience. He's the same way with people in general, and he'll be wary toward them until he feels he has the measure of them. This can take anywhere from a few minutes to a few years, depending on circumstances and the person in question. This is only basic trust, though. He very, very rarely "opens up" to people and generally dislikes when someone tries to get close on a personal level. If the Chief trusts you he's fairly sure you will not do anything (out of will, ignorance, or incompetence) to get him killed, but he will most likely not tell you about his life or confide his problems in you.
His gullibility comes in that while he may not trust people initially, once he's evaluated them as "safe" his guard tends to go down quite a bit and it's easy to string him along. The prime example of this is 343 Guilty Spark. Spark helped the Chief fight the Flood and was not outwardly hostile toward him (at least initially), so the Chief very nearly fired the first Halo because he had decided it was safe to follow Spark's instructions. He was alone at the time, so he also had no one else's suspicion to influence his evaluation. If it hadn't been for Cortana bursting in and stopping him, he would have done it. As long as what you're telling him seems to make sense based on the information he has (all the better if it's information you've given him and time is short), the Chief is likely to believe you.
John and Kids:
The Chief's a little weird with kids, mostly because his own childhood was ripped violently away from him when he was a boy. They make him a touch uncomfortable because he's not sure what to do with them or what's expected of him. He's better with older children than with younger, and tends to treat them according to how they act. If you behave like an adult, the Chief will (mostly) treat you like one in turn. He will soften things for those he sees as more fragile, and be blunt and straight with the ones who've proven their maturity to him.
Perhaps surprisingly to those who do not know his background (read: almost everybody) the Chief has a good deal of confidence in the capabilities of kids with training. He himself was able to kill a man by eight and fighting in full on warfare against the Covenant at fourteen, so his view of this is a bit skewed.
He has a tendency to look out for younger people, though he's subtle about it. This is especially true in the case of those who have been flung into strange circumstances like his own. He sees something of himself in them. If called upon, John could likely easily take up the mantle as a mentor as Chief Mendez was to him. He is a firm but fair teacher. He will work you to the bone, but so help him you will learn something from it. And if it's something that keeps you alive, then he has done what he set out to do by it.
John and Relationships:
John is not close to people. This is a fact of life. Fire burns, water is wet, and the Master Chief is as much of an island as he can be. There are multiple reasons for this, one of the simplest and most obvious being that he has lost many people in his life and it's easier to get past if he doesn't have a strong bond to them. Another easy one is that his origins are so very, very classified. He takes no pleasure in lying, so it's just easier to avoid the questions altogether by keeping people at arm's length.
John's leadership position also isolates him, as he sees it as his responsibility to maintain his composure at all times for the benefit of everyone else. If people know him well, it's a possibility they'll be able to spot his own distress and it's not good for anyone's peace of mind. John's problems are in his mind his and his alone, and he will not readily share what he sees as his burden. He will tear himself apart inside, if it comes to it, to protect those in his charge.
This tendency toward self sacrifice is part of another reason why the Chief does not form close ties. It's not actively seeking it so much as that he sees himself as an acceptable loss in the line of duty. He has a job to do and he will pour absolutely everything he has into it. The Chief's life isn't his to promise to anybody, and he hates making promises he knows he can't keep. He belongs to the UNSC, he belongs to his superiors, and he belongs to his Spartans. He has a lot of responsibilities and there's little room for anything else. This especially factors into his unwillingness to have any romantic connections. He feels he'd be unable to provide for whoever it may be, be it in security, time, emotional openness, or shelter from the mess of his life they'd be inevitably pulled into. It's hard for him to even understand why someone would want to be with him, if they truly knew him. Couple that with the fact that he has so many things to hide already and so many barriers and how his continued life is never guaranteed, and it strikes him as a shamefully selfish action doomed to failure from the start. That way lies only heartbreak, and John knows it. And it's one of the few things that actually scares him.
Better he be alone on that inevitable day he's killed, and not pull anyone down with him.