And if so, then would that right extend to the right to choose the timing and method of our death?
The concept "right" implies choice, so the right to free speech is not an obligation to speak, the right to own property is not an obligation to own property. The right to live, thus, is the right to live if you so chose, from which it follows that you also have the right to chose not to live. You don't have a generic right to die only in the manner and time that you chose (which would imply that you have a right to whatever is necessary in order to keep you alive until the time that you've selected for your demise), but, if you chose to end your life at some time in some manner, and you can stay alive until that time and can implement your death, they you have the right to act on that choice. (Modulo respect for other people's rights, i.e. you don't have a right to die in a nuclear holocaust that wipes out NYC).
However, in a State Boon theory of rights, where rights are all and only the actions deemed by the sovereign to be "rightful", then the only way to answer the question is see whether a right to die has been granted.
Rights are a silly way to talk about this kind of thing.
Does the pilot of the plane you are currently on have the right to die? Well, if someone shoots him, we cannot punish him for that, but if he just up and decides to die and leave us unable to land, that is not his right. If it would be illegal to prevent him from doing this, the world would be a worse place.
We can never punish those who die against our will, so there is no power to oppress them. Nor can we simply allow people to die at will. People have obligations, and we are not in a place to lift those obligations without considering all of the other people affected.
So the entire framing of 'rights' is inappropriate here.
We can consider whether we should act against those who attempt to die. The question is whether they generally genuinely wish to die? And the answer is generally no. Some people attempt suicide a second, or a fifth or a hundred-and-first time, but suicidality is usually transient. Those saved from suicide are generally grateful eventually. So generally we should interfere in suicides.
The strongest proof, to my mind, is the frequency of post-partum depression. The mother of a newborn has no right to deprive her child of access to her without adequate planning. But after pregnancy is a common time to have suicidal thoughts. She is meant to be rescued. It seems very likely that being suicidal at all is a naturally evolved test of whether she has social support, or whether someone else would be a better mother to her child. If we save her, we are there, and her child is safer for knowing that. If we don't, we are not there, we are not a good community, and the child has a second chance at finding a better one.
The possibility of interference itself is a good predictor of whether the attempted suicide was genuine. People who actually want to survive generally choose varieties of methods that have safety hatches. Drugs fail if there are people who will find you before they fully take effect, so attempting suicide by medication is often a test of whether one has actual social connections who will notice your absence. People often stage the attempt at a time that provides a good test of their perceived value, when they are likely to be found. People who are more certain they want to die use decisive and violent means.
But ultimately suicide cannot be prevented. If clarity is tested in some rigorous way, there is no reason to force people who genuinely want to die, to do so unpleasantly.
At the same time, if I want to die by being drowned in fudge, that is a horrible waste of fudge, so the idea that folks have the right to choose the manner of their death is bizarre. A chosen death is not a sacrifice that earns one special treatment. We can let you go, but in some way that is convenient and non-traumatic for us.
There are now, I think, two states that allow suicide, after psychiatric review by high doses of slow-acting benzodiazepines. So transience of the wish is tested professionally, but the death itself is not unpleasant. And it is kept from being unpleasant in a way that does not make it short. So we are still able to prevent it if the person dying indicates a late change of mind.
This is a nice compromise between a "right to death and its circumstances" and an "obligation to save everyone" even when it cannot do them any good.
Here's an alternative way of looking at it. Every right has a corresponding duty. The (prima facie) right I have to express my opinions, e.g., consists in the obligation that others have not to prevent me from expressing my opinions.
So do people have an obligation not to interfere with your choice to die? Obviously it depends on your decision making capacity, whether the choice to die is a fully informed decision, the likely consequences of this decision, etc. The exact same applies to the right to life (or the obligation others have not to interfere with your continued existence). Perhaps these two rights seem asymmetrical because most people don't have to think twice about whether they want to exercise their right to life. But this surely is not true of all people.