We, each of us individually, feel a sense of self worth, a sense that something of great value will be lost if we cease to be.
Some of us do. Some of us have a more moderate sense of value, and many of us feel as though we have very little value, if any at all. For many of us, our sense of value comes not from a sense of ourselves as valuable to ourselves, but as valuable to others, especially if others depend upon us. When we are isolated, we can easily lose this sense of value, and lose our sense of self worth in turn.
Does committing suicide imply that a person does not have his own sense of self-worth?
Not necessarily. Whilst a lack of a sense of self-worth is probably integral to the decision of many to kill themselves, it is also quite conceivable that a person can have a sense of self worth but feel that no-one else shares this sense; that their value is unacknowledged by others. They may feel misunderstood, neglected, perhaps unfairly disparaged. A sense of being unjustly treated, ignored and/or misunderstood can prove extremely detrimental to mental wellbeing. Combine this sense of neglect with a sense of isolation, frustration and/or resentment and the act of suicide can become an act of protest; a cry of "why can't you see me for who I am?", or "why do you ignore me? I have a lot to give! Why should I persevere?".
Is a suicide someone who simply lacks the sense that he is an intrinsically valuable part of the universe, or is it instead an act of deliberate vandalism? Does he understand that he is destroying something of great value and chooses to do so?
I'm not sure I have the sense that I am an 'intrinsically valuable part of the universe'. I make this personal statement not to invite sympathy (I feel immensely lucky), but to express that it is quite possible for a person to not hold to 'intrinsic value' without feeling suicidal. The converse is also true; that it is entirely possible to feel as though one is an intrinsically valuable part of the universe, and to be suicidal nonetheless, as mentioned earlier.
As for vandalism, many people contemplating suicide might share a similar sentiment; a destructive impulse which again belongs to the realms of protest, anger and abdication. Note though that the act of suicide can be the result of neither proposition; ie. that a person can have a sense of value and kill themselves but do so in a way that is not motivated by vandalism, but by other reasons including exhaustion, loneliness, pain, insanity and fear.
It is sometimes assumed by people (and I'm not suggesting you're doing this), that the person who commits suicide is a priori irrational, but this is far from the case. Suicidality, in many circumstances, is a very rational, comprehensible (if tragic) impulse, and acknowledging this fact is very important when dealing with the suicidal, because to label them irrational is to in a way deny their tremendous suffering and may contribute to any sense they have of being misunderstood.
Does a suicide understand what he is doing?
There is no one-size fits all answer to this. Some people kill themselves in the throes of delusion. Others have a severely (medically) inhibited sense of themselves thanks to altered serotonin levels in their brains. Others have even been emotionally manipulated into suicide. Many kill themselves whilst under the influence of mind-altering substances which augment negative emotions and /or reduce inhibition. But it must be acknowledged that many do so out of sheer desperation, out of an inability to solve the problems that plague them. Many of these people in turn do it knowing precisely what they're doing - including harming those who care for them - but because they see no other option; no other means of escape.