Good Question. Definitely not.
I might use them if I like them. But I won't consider them "core". I think the definition of core is going to be a lot more subjective in 4E.
"core" is just a word. It can reasonably be used to refer either to the most basic books, the ones you're supposed to have to play the game at all, or to the books that apply to the game in general instead of to specific settings - the "core" that the settings all revolve around. We're just more used to the former meaning than the latter.
That said, I would've preferred to keep using "core" as I'm used to, especially since I use that term all the time for other RPGs where the second meaning is inapplicable, but if WotC sticks to their guns on this the second meaning may eventually drown out the first. Either way, I hope it gets resolved fairly soon; I hate ambiguous communication.
Then they will become "The Number Books."
The idea I think is to make a clear distinction between the first three and anything subsequent. That way there is a designation that says "This is not required for play, no matter how much the publisher wants to imply it is."
No. The core by defnition, is what is core to the game, the rules that everything else will run off. And I'd hope what is core to most games is included in the PHB, DMG, MM.
PHBII, DMG II, MMII and the books that follow those I'll view in the same light as Unearthed Arcana, Complete Arcane, PHB II (3.5) etcetera.