If a Dragonfire adept meets the stat requirements for a Metabreath feat from Draconomicon, can he still take them even though his breath weapon technically doesn't use a cooldown? Would he just be unable to use his breath weapon for the specified number of rounds, or would it only apply to the breath effect he used for the metabreath effect?
Only with a House Rule That Permits It
Because the dragonfire adept supernatural ability breath weapon says that "you gain a breath weapon that you can use at will as a standard action," and Metabreath Feats in the Draconomicon says, "To take a metabreath feat, a creature must have a breath weapon whose time between breaths is expressed in rounds" (66), the answer is no, a dragonfire adept can't take metabreath feats.
Although there's an argument that the time between a dragonfire adept's breath weapons is 0 rounds--or even 1 round using the breath effect ability--, that duration is never explicitly expressed that way, instead using this language: "Also, you can't apply the same breath effect to your breath weapon in two consecutive rounds (though you still can use your normal fire breath weapon every round)."
Seriously, only the DM knows how an effect functions in a game wherein the DM permits abilities to be employed by creatures that specifically can't employ them.
A Somewhat Murky Workaround
This might not be the real question, though. If the real question is, "Is there a way for a dragonfire adept take metabreath feats despite his breath weapon's lack of a recharge rate measured in rounds?" then I point you to general feat Power Surge (Dragon #313 57) and janusjones's Dragonfire Adept Handbook, which discusses how to do exactly that and many more dragonfire adept features in much greater detail.