Your questions have a few answers depending on the situation. It sounds like the simulations you're describing are sort of "pre-code" or "not actively arresting yet but probably about to" sims where your instructors are prepping you for responding to a patient telling you something is wrong. If a patient says they feel like they are going to die ALWAYS do a thorough reassessment as some events give a pt a "sense of impending doom" (STEMI's, AFib with RVR, some strokes, etc). By reassesment I'm talking EKG, troponin or lactate if possible, thorough head to toe, vitals. Call the MD with results or call sooner if you get abnormal findings. In these scenarios where the pt isn't in acute distress you can say "what makes you feel that way", "are you feeling bad", etc as part of the assessment.
Now, if you're actively treating and working up a patient who might actually be about to die and they frantically look at you and ask you that question, I usually tell a soft version of the truth. Everyone will have a different answer on this one, some people will lie, some will try to be thereapeutic and ask "what makes you think that?" or "tell me your concerns". I think in an emergency scenario these are not great responses because the pts answer will be "I think I'm going to die because y'all are doing stuff really fast and using words I don't know and bringing strange machines in here and I have 2 more iv's and 6 more people in my room than 15 minutes ago". I usually say something like "you're very sick right now because your heart is in a dangerous rhythm" (or you're not getting enough O2 or whatever the reason is), "a lot of things are going to be happening very quickly, tell me if you feel strange in any way, I'll be right here" and then I make sure I AM right there. You don't want your pt to panic, but if it were me I wouldn't want to be lied to or have someone ask me my "concerns" when it's obvious something emergent is happening. Just try to stay calm and keep talking and being with to your pt.
As far as the question about people who are stable at the time saying they're going to die and then actually dying days later I will just say there are definitely a larger number of people who say they're going to die and DON'T die. Go to an ER and everyone there is dying from their ailment, at least that's what they say. Most don't die from said ailment. Still do a thorough reassessment, always.
I'm not aware of a proven physiological response where people feel calm days or hours before they die, unless it's a hospice situation or they're on anxiolytics something. However, there is a theory that a chemical called dimethyltryptamine is released at the time of death (as in heart stops or lungs stop) that some say can give a sense of calm. I'm not sure on the validity of that but a lot of personal accounts tell a similar story.
TLDR: it depends on the situation, reassess the patient always, there is no "right" answer.