There is one good trial and it found an effect. Dengate and Ruben (2002) took 27 children whose behaviour had already improved on an additive-free elimination diet, then challenged them double-blind with calcium propionate hidden in ordinary bread. Significantly more children worsened on parent-rated irritability, restlessness, inattention and sleep than improved. It is a small study in an already-sensitive group, so it does not mean the preservative affects every child, but it is a good deal more evidence than most additive complaints have. The effect built over several days, which is why a one-day test will miss it.

It is regularly reported in elimination-diet practice, where propionate is removed alongside salicylates, amines and glutamates, and people do report headaches returning on rechallenge. But there is no controlled human trial on migraine specifically, so the honest position is that the headache link is anecdotal while the behavioural link is trialled. If you suspect it, the fair test is two weeks of bread without E282 followed by a deliberate rechallenge over several days, since a single exposure is unlikely to show anything.

Propionic acid genuinely is a natural product. Propionibacterium is what puts the nutty tang and the holes in Emmental and Swiss, and your own gut bacteria make propionate constantly from fermented fibre. That does not automatically make the added version harmless at the dose it is used, and it does not make it irrelevant to a sensitivity log. Naturally occurring and well tolerated are separate questions.

Yes on both. The calcium comes from a mineral salt, not from milk, and nothing in its manufacture is animal derived. It is also gluten free in itself, though it is almost always found in wheat bread, so the product around it usually is not.