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From what I understand the answer is no. You are not finished leveling up until you apply everything. Here is the section of the rules:
When adding new levels of an existing class or adding levels of a new class (see Multiclassing, below), make sure to take the following steps in order. First, select your new class level. You must be able to qualify for this level before any of the following adjustments are made. Second, apply any ability score increases due to gaining a level. Third, integrate all of the level's class abilities and then roll for additional hit points. Finally, add new skills and feats. For more information on when you gain new feats and ability score increases, see Table: Character Advancement and Level-Dependent Bonuses.
I have seen many people use houserules to change this, but, at least from my understanding, RAW is no.
By the rules, no. Personally, I would allow anyone to take a feat that they don't qualify for. They just wouldn't be allowed to use it until they met the prerequisites.
