3 year olds will ask questions about anything they see around them. If someone anywhere was talking about death -- a pet, a grandparent, something on TV, she'll ask about it. That part isn't so unusual, except that it feeds into what else is going on, and stresses you out.
The rapid pulse rate may have to do with her own fear about what's going on. She's not in control at those times, and she can't stop it, or block it out, and it's scary. Plus it scares Mom, so that makes it even scarier.
If it is vertigo, why don't you try a wee bit of peroxide on a q-tip and just drip it into the ear canal ? Don't stick the q-tip inside the canal, just enough to push on the side and drip some peroxide gently in. If there's water bubbles in there, ithe peroxide will break the surface tension of the water, and get the ear fluid moving better. Another solution for dizziness, is to buy "sea bands" -- they are little terry bracelets that are about an inch wide, with a bubble on the inside. The bubble goes on the pulse spot, and they help people with seassickness. My mom had a dizziness problem that caused nausea, and that sometimes helped. (What actually solved that problem was to sit sideways on the bed and throw herself back, which jarred the ear fluid and caused it to flow and stopped the problem entirely, but I don't recommend tossing your daughter around given her former head trauma)
Other things that could cause dizziness -- blood sugar. Does she feel at all "shakey" before this happens ? If her blood sugar is low, then she could become dizzy.. . . . This came to me when you said it happens when she gets out of bed, and at mealtimes. Not so much AFTER eating, but if it's BEFORE or during, it could be a blood sugar issue. The food pyramid thing doesn't work well if someone has low blood sugar. They need more protein. Try feeding her eggs for breakfast for a week, and see if it helps. (Protein tends to help your body establish a floor for blood sugar for the day). And be sure she gets some good protein at supper.
Also, when she sees circles, if she puts her head below her knees, does it go away ? If she isn't getting oxygen to the brain, getting her brain lower will help. This would make sense with it happening in the morning -- because she's getting up from a reclining position, and maybe her BP doesn't jump out of bed as quickly as her body does. I would also consider trying to add more iron to her diet. Tuna, steak, eggs. The iron in our blood helps to carry the oxygen around, and blacking out is usually a lack of oxygen to the brain. She isn't quite blacking out, but it sounds like she's recovering after "almost" getting there.
You know how you have those "falling" dreams, and you wake up and you're scared to death ? I suspect that's what it's like, and the problem is that it's happening while she's awake. Very very scary. So I would tend to assume the rapid pulse is the scary factor. Rapid pulse and quick shorter breaths are panic responses. (not sure about the pupils)
I guess, after all this keyboard stream of consciousness stuff, I would say to start going with eggs for breakfast, be sure she gets meat of some kind into her for supper and maybe give her a small snack an hour before supper. See if changing her eating habits solves the problem.
Explain to her that when she gets dizzy, one way to solve it is to get her head lower than her heart. She can lie down, or she can bend over in her chair and put her head between her knees. Come up with a code, for her to use to tell you it's happening, and then she should bend over if in a chair (not enough to fall out -- her feet don't go to the ground yet) or to get down on the floor so she won't fall. And as Mom, you should show up, and rub her back, talk to her and comfort her. See if the circles go away when she gets her head down. If you can come up with a response to the problem that works, and helps to allay the panic, then she will be able to cope, and the rapid pulse stuff and fear will go away. If she doesn't panic, but her pulse is shallow and quick, then there's a more serious issue going on.
I'm really glad you've found a doc who is taking this seriously, and is paying attention to the problem. So many times you can feel as if they blow you off when they aren't sure what is happening. This doc is engaging his mind and his sense of wonder, and is asking you to produce more data, so he can get his scientific mind around the problem, and help. It may not help you deal with it "right now", but he's going in the right direction.
I had a few really wierd incidences, which I attribute to menopause, when I was simply sitting in a chair reading a book and thought I was going to pass out. It was freaky, and the second time, I called my hub who took me to the doc, and that petrified me, so when we got there my BP was way up, and my heart rate was, too. After talking with the doc, he checked my heart rate and BP again, and it was back to normal. So they ran me thru all kinds of tests, and came up with nothing, but a friend of mine later told me the EXACT same thing happened to her, and her doc put her thru all the same tests, then told her it was bodily stress caused by the hormones changing in menopause, and it was OKAY. The only thing we found "wrong" with me, with all the testing, was low blood sugar. So now I try to avoid sugar, I eat protein, I eat Balance Bars because they have protein in them, and the carbs are slow-metabolizing so they don't spike the sugar up and then down, and I am able to avoid any low blood sugar symptoms for 6 months or more at a time. And I haven't had a recurrance. But I also learned that sometimes low blood sugar and a mitral valve problem can go hand in hand -- which makes me wonder if my sugar went down, the vitral valve stopped working correctly, and then when the panic set it, everything got moving again and it passed. . . ??? . . . so, part of what I'm thinking, is that if it is low blood sugar, then it could also be affecting her circulation, and it IS really scary when that happens. Also, if it's "stress" related, that doesnt' mean it's stress due to emotional things -- there could be a stress reaction going on inside her, not necessarily hormonal, but something may not be 'quite right' and it could cause a bodily stress thing to happen. Could be blood chemistry, could be almost anything.
Please be sure to report back when you find anything -- or not. This has got to be tough for you, and I imagine everyone else is feeling a lot like me -- wishing they could solve it for you; she's young to have such scary problems, and finding the cause is hard, also, when children are little. Hang in. Hopefully someone else has been thru this and has some more solid advice than mine, which is really just "ideas". :-)