[Excerpt from a mail, concerning why the Arryn moon-and-falcon is so strangely stylized and the Hightower's of Oldtown]
-- Arryn. That's complicated. It has to do with a cover that you likely never saw; the =first= cover of the US hardcover, a painting by Stephen Youell that appeared on the ARCs but was scrapped in favor of the silver foil before publication (though it was used for the Swedish edition, of all things, so maybe you =have= seen it, if you have run into the Swedish edition). The scene it illustrates is Catelyn's meeting with Ser Brynden at the Bloody Gate. In the painting, Brynden is carrying a staff with an Arryn banner on it. My text had mentioned the Arryn "moon-and-falcon" but given no details, so Youell came up with the sigil as you see it. Then later, when we decided to put the sigils in the appendix, Virginia Norey did the designs... but on Arryn, she felt compelled to stick with Youell's version, since it was already "canon." By the time it was decided not to use the Youell cover, the Norey drawings were already done, so there we were. Which is a long way of saying... no, the Arryn sigil is not at all what I had originally imagined. When I was writing the book, in my head I saw a sky blue field with a large round full moon in its center, and a blue falcon soaring upward outlined against it. However, having appeared in the appendices of two books, and on the cover of the Swedish edition, the current version would seem to be "it."
As for the broader questions you raised...
The Hightowers can be legitimately referred to as being "of Hightower" or "of Oldtown," either one. Hightower is their castle/keep, the tallest structure in the Seven Kingdoms, and one of the oldest, a massive stepped tower with a great beacon on top, to show ships the way to port... kind of like the Pharos of Alexandria, but larger, an inhabited castle as well as a lighthouse. It stands in the center of Oldtown; the city grew up around it. And Oldtown =is= old, thousands of years old as opposed to King's Landing, which is only three hundred. Until Aegon's coming, it was the major city of Westeros. The Hightowers are one of the oldest families in the Seven Kingdoms.
You will learn more about the maesters and the Citadel in future books.