Well first off, congratulations on your baby.
I've never been pregnant, but it causes a LOT of changes in your body, and with all the extra hormones you're probably bound for some extra anxiety as well. Now, I'll tell you what I tell everyone--it's not in anyone's interest, least of all your doctor's, to lie about the health of your heart. You could sue them into the stone age if something was actually wrong with you that they failed to diagnose, and they know that.
Panic attacks can cause your heart to look wonky on some tests. But if they ran extra tests and nothing came of them, you're fine. You're not going to be 31 and suddenly drop dead of a congenital heart defect. Those usually surface much earlier in life.
Heart disease does not exist in absence of all other symptoms, no. You usually have blood pressure or cholesterol issues at least. At 31, if nobody in your family has spontaneously died young of heart problems, the chances of you doing so are ridiculously remote. You don't smoke, which is the BEST thing you can do when you're worried about your heart. You're at a normal weight for your height, your blood pressure is fine. What you have is an anxiety and panic disorder, not a heart condition. You're having palpitations, which causes you to be hyper-aware of your heartbeat. Anxiety is the biggest cause of those, and then they cause your anxiety, so it's a cycle. Panic attacks usually strike me when I'm asleep too, I hate that. If you notice them more when you're on one side vs. the other, that may be because at some point you heart the (untrue) myth that sleeping on your left side can cause heart problems. I know it plagues me even though I know it's completely unfounded. And now that you're pregnant, you're entering into the wonderful world of acid reflux, which is going to be a burning pain in your chest and throat, especially when lying in certain positions at night after eating. You may feel that and start to get nervous, which means more palpitations.
I suggest you speak to your doctor about your anxiety issues if you haven't already. You may be out of luck on medications because I don't know what's safe to take, especially this early in your pregnancy, but there may be options. Therapy may be a really good idea for you, learning what's at the root of this fear about your heart and how to deal with the cycling thoughts and the physical symptoms they cause. What you need to do right now is learn to relax and deal with these anxiety attacks, because the extra stress isn't good for you or the baby. Believe me, it will stress you out enough once it's born. Talk to your doctor and see if you can find a therapist that specializes in cognitive behavior therapy and panic disorders. You might find some real help there, and there's no side effects like with meds.
Short answer, no, this pregnancy is not going to overstress your heart and kill you, although you may wish it had once you hit labor, from what I've heard.

No, you're going to be fine, but you need to find someone to speak with about this panic, because it's not doing you any favors and this baby does not need an anxious mother right off the bat. Find some help for this, because this is extra stress you don't need.