"Evasion" is the lowering of your degree of (conceptual) awareness. To evade is to not make yourself aware of the various connections between everything you know; to ignore or deny what you know. Example: "I know that smoking causes cancer, but I will ignore it". Evasion is more than merely ignoring facts: you are also evading when you believe something in lack of evidence or, worse, despite the evidence. Example: "I believe in God, not because there are any reasons to believe in God, but because I want to believe in God". Observe that in both cases you are putting your emotions above the facts of reality.
Failing to make connections, due to intellectual passivity, is not necessarily the same as evading. It is drifting. Example: You become aware of a pattern in the economy (price inflation and the Fed printing more money) but you simply do not bother to connect it to what you know about economics and everything else that is relevant.
Context-dropping is, as John Paquette said, what you end up with when you fail to connect some relevant knowledge. Context-dropping is not necessarily done on purpose (in which case it is an evasion). It can be the result of failing to make a connection because of a mistake. Or maybe you are simply ignorant of the relevant facts. It can also be the result of having a bad habit of not making connecting, i.e., of drifting too much.
Any version of the broken window fallacy is an example of context-dropping. Example: You argue that government stimulus would be good for the economy, because it would create new jobs, but you fail to connect the fact that for each stimulus dollar the public sector consumes, there will be one dollar less for the private sector consumes. This policy will, therefore, not create any new jobs. At best it will only create some jobs in one sector at the expense of another. (In fact, since the precondition for any economic growth is savings, this policy of government consumption actually undermines the creation of more wealth and jobs in the future.)
Now, what is a "rationalization"? A "rationalization" is an attempt to cover up your emotions with arguments. The purpose is to make your emotions seem rational. Example: "Why should I not download this song? I will never buy it anyway, so what's the harm?" (See any random teenager who illegally downloads music.) Or: "I want to believe in a God, because it would make me feel better, therefore I will formulate this elaborate philosophy to justify the belief in God." (See here, for instance, William James or, for that matter, Immanuel Kant). Or: "I don't hate the rich, but they have to pay higher taxes to balance the budget!" (Why not just cut government spending instead? See, for example, Mr. Barack Obama.)
I think this is best answered by visiting http://aynrandlexicon.com and looking up these terms.
In general, though, "evasion" denotes any volitional act of avoiding awareness.
Context dropping is a cognitive error, a failure to relate a fact to its necessary context, which might be intentional (in which case it is a form of evasion).
Rationalization is, as Carl says, the act of faking rationality -- of whitewashing the fact that you are making choices based on your emotions.
I believe that context dropping can be an innocent error, like forgetting, while rationalization cannot be.