ANSWER: I would say, candidly, that your analysis is largely incorrect. Let me try to explain in a few steps. First, where an agency is "placed" is not really the issue. Rather, the issue is whether the agency is involved in the execution or implementation of federal law. Any agency so doing is part of the "executive branch," at least in a sense. Now, some people will say that independent agencies are not part of the Executive Branch because they are independent of the President--independent typically because their heads are not removable by the President at will. I would say that this lessens presidential control, but it does nothing to obviate the underlying reality that the agency is executing federal law. Second, sure, the agency is "quasi legislative" in the sense you describe. But so is every agency. All execution of law involves some discretion, and often it is sensible to announce in advance, in the form of regulations (that look like statutes) how the agency plans to exercise that discretion. But this does not make them an extension of Congress, or part of the legislative branch. Recall that, under cases such as Chadha and Bowsher and Buckley that Congress can play no role in the execution of federal law. So this agency cannot constitutionally be a part of the legislative branch. Third, once we resolve that the agency is part of the bureaucracy that is executing federal law (whatever exact label we place on it), it should become clear that there can be no line-item veto problem. the President is merely adjusting how the law is being executed; he is not altering the statute itself. Finally, the delegation issue is only relevant in evaluating the standard that Congress has given the executive branch as a whole. So long as that states an intelligible principle for enforcement, that doctrine is satisfied.
QUESTION: Also, what are agencies like the EPA actually classified under and who is in charge of them? It seems that they are legislative in the fact that they create regulations but also executive as they enforce these regulations (through fines and whatnot). So is the legislative branch or he executive branch in charge of them or are they independent from both?
ANSWER: The EPA has responsibilities of enforcing and executing federal law, thus it is a part of the Executive Branch. The head is the Administrator of the EPA. Again, you are right that much of what it does seems legislative in nature. But the better way to understand this is as the exercise of discretion in the enforcement of federal law.