Re: Did UK make ever something positive to make EU stronger?
While you're correct Schengen is only about removing border controls *for people only*, it helped anyway a lot to move across Europe with less issues. It really had no real impact on the refugees crisis. It was just the scapegoat. The real problem is how to protect the *external EU borders* of those countries under refugees pressure. If Britain would have been in the place of Greece or Sicily, it would have had the same problems regardless of Schengen. Just look at the US and the Mexican border.
No EU leader took a stance about how to really control immigration across the Mediterranean. They all, including British ones, just hoped it could stop by itself, or the most exposed countries would have been forced to sustain all the issues. A pure "mors tua, vita mea" attitude. Just, some are able to reach the Channel as well... just as you can't sink them at sea, you can't really fire at them later. Nor you can't believe to jail them all.
It was far better some countries like Italy couldn't control their currencies. They have far deeper problems (huge debt, high gov expenses to buy consensus, corruption, high taxes and evasion, low productivity, organized crime) than the value of the currency. Depreciating their money would have been just a cheap short term solution (useful for politician who can't see past the next election, and some exporting companies), which would have *increased* most people problems long-term, and lead to more emigration from those same countries.
Meanwhile, a lot of British immigration comes from countries *outside* the EU. From the ex-empire, for example. Will it stop as soon as British leaves the EU? No, Indians, Pakistani, Africans, etc will still arrive. Most of them arrive sitting on a plane, with a valid visa. Good luck, with them.