OK, I'd like to address a very specific point: genuine refugees from war-torn/collapsing countries. If you take harsher line against the other kind of migrants, is is justificable for rich and stable Europe to turn her back on them? Let them f**king die or drown?
Whom are you asking?
I can only speak for myself.
For individually persecuted people there should be the possibility to apply for asylum in the EU from both inside or outside. For these people there should be no upper limits (at least until now the numbers have never been high anyways).
For refugees from war-torn or collapsing countries there should be contingents for both temporary and permanent resettlement. Participants would generally apply for these programs from outside the EU. I think that we have the duty to take our share, but only within certain limits.
Deporting people who entered the EU irregularly would be the most effective mean to prevent that more of them drown or enter dangerous countries like Libya.
I would add that many people from different political backgrounds put refugees and immigrants in the same bag, for all kinds of motivations.
When there is a huge refugee crisis like in the Syrian case, some of the smaller neighbors (Lebanon, Jordan) will get into difficulties, and that has nothing to do with "should do", they are already doing as much as they can. In my opinion (probably not representative) allowing a certain contingent of vetted refugees to come to e.g. Germany in a controlled manner would have been a good decision. But then, after we have taken our share, we cannot and should not do it all alone. (I'm not saying that e.g. Germany is the only country that has done something, look at Lebanon, Jordan, but also Sweden, Greece, Austria...) Finally I think that we should not force other countries to take in refugee contingents.