Generally you are expected to show the same passport on entry and exit to Immigration authorities (that's in countries that have exit controls, of course, but they're the majority anyway). You'd have to have a good reason to use another passport to exit than the one you used to enter.
It did happen to me once: I have two passports, and grabbed the wrong one when leaving – it was a bit of a hassle to pass through Immigration. I had to provide a bunch of information as to why, the original passport's number, sign a declaration, etc... Not worth it, unless I had no choice.
As for the question about a union – in the case of the EU it's probably fine. You're not stamped in anyway, and as long as the ID matches the person, what the officer sees is an EU citizen. Move along!
Connecting flights: sure, you can do that, possibly. I've done it often: one of my passports had plenty of pages left, the other not so much. So I'd present the one with more pages, regardless of which one I used to leave.
Finally, about disclosures. Some countries share information with other countries, but generally, they don't need to. I was asked by a Malaysian Immigration officer why I had 2 passports, and was not using the same one every time I visited Malaysia. She insisted that I should not do that. She knew that because she had the record of my entries and exits on her screen, and the previous entry had a passport number that was different from the one I had given her, and it was still valid – so she assumed, rightly, that I was using 2 passports. She wasn't happy, but who cares? Bottom line, they might not know for sure, they they have lots of data to have an opinion on the matter.