> And if you think I'm factually wrong, I'd love to see crime stats demonstrating this.
Sure. Only recently, the German BKA released numbers [1] on crimes committed by immigrants. As you can see, until September last year, the number of immigrants (the red line in the first graph) increased faster than the number of crimes committed by immigrants (the green line in the same graph). Generally, immigrant criminality is dominated by petty offenses (shoplifting, fare evasion) and immigration-related offenses (such as forged passports). As far as nationality is concerned, immigrants from Balkan countries dominate the lists of suspects (on a per capita basis), while crime among Iraqis and (especially) Syrians appears to be disproportionately low.
Note that this does not mean that immigrants do not commit less crimes than residents. Low socio-economic status alone means a higher propensity to commit crimes (and poorly integrated, low-income immigrants have been a concern since forever, whether that was Turkish guest workers [2] or the Russian-German immigrants of the 1990s [3]). But there is no evidence that Muslim immigrants are more likely to commit crime, either. In fact, technically the (predominantly Christian [4]) Balkan countries seem to be far more likely to turn out criminal immigrants.
Note further that most of the suspects in Cologne seem to belong to or be affiliated with pre-existing crime groups [5] from Morocco [6] and Algeria, people who were either there illegally or were using the refugee crisis as a cover. While these should be deported, this is difficult, because their countries refuse to take them back.
In general, traditional criminogenic factors seem to be plenty sufficient to explain immigrant crime, and that's really not all that surprising. Whereas ethnic and religious factors seem to be either sourced from anecdotes or the result of incidental correlation.
Completely separate from that is the challenge of integrating such a huge number of immigrants into German society. Not to put too fine a point on it, but that had been botched twice before; there is the language barrier, widespread skill gaps, and the difficulty of cultural acclimatization to deal with.
> The establishment left in Europe, from what I can see, is lying about the problems of Muslim immigrants and crime/other social problems.
If that hypothesis of yours were correct, then in Germany, this would not just be the establishment left, but also plenty of establishment conservatives. While politicians did botch the response to New Year Eve's events, the simpler explanation is that they were worried about attacks on refugees by the far right (which also promptly happened). While not exactly pogroms, far right violence has also been a persistent concern and many people still remember Solingen [7] and Rostock-Lichtenhagen [8].
I don't see anybody lying or otherwise dissembling about the problems related to integration; it is no secret that previous immigration waves were far from successful in this regard (as noted above), and nobody expects this to be a cakewalk.
[1] http://www.spiegel.de/panorama/justiz/fluechtlingskrise-bka-...
[2] Recommended reading on the situation of Turkish guest workers until the 1980s is "Lowest of the Low" ("Ganz unten") by Günther Wallraff.
[3] Poor integration led to a crime wave among young white Russians and probably helped the Russian mafia strengthen their foothold in Germany.
[4] Which, of course, has probably nothing to do with religion or nationality, either, but more with organized crime originating in Eastern Europe.
[5] http://www1.wdr.de/themen/aktuell/vorfaelle-hauptbahnhof-koe...
[6] Morocco in particular encourages emigration to Europe; it reduces unemployment and results in more cash transfers to Morocco. However, because Morocco is far from a free country, asylum requests still have to be taken seriously.
[7] https://en.wikipedia.org/wiki/Solingen_arson_attack_of_1993
[8] https://en.wikipedia.org/wiki/Rostock-Lichtenhagen_riots