A bit of history.
There was a terrible scandal in the late 1990's when it was uncovered that the French police were briefly detaining illegal immigrants to France and then shipping them on to the UK so that France wouldn't have to deal with them. They were actually sneaking them into the backs of trucks and lorries and ferries and what-not just to be rid of them, and the UK was taking the brunt of it.
When this was uncovered, there was a *HORRIBLE* row with France (and properly so to my mind). As a result, the asylum rules were changed so that a claimant could only claim in the first EU country he entered. This caused lots of claimants to "lose" their identity documents before arriving in the UK, but that goes on to a different story.
The other result was an acceleration of the juxtaposed controls referred to by John. They have been largely successful.
To your other question, the UK is often targeted because of the housing and income support and health care benefits. Also because many people have at least some exposure to the English language.
To your last question, those caught by our juxtaposed controls are turned over to the local authorities who then release them unless they have committed a custodial offense.
Based upon my own experience, I think the French authorities are still assisting some to get to the UK, but that is a massively informal viewpoint.