I do not want to focus so much on the linked article, suffice to say that EXPRESS is using very lurid language in the article. Instead I focus on your three specific claims.
The primary source is the official police statistics. The statistic itself gives no context or explanation. It has tables listing the cases that became known (bekanntgewordene Fälle), of those, attempts , clearance ratio per month.
Tables are for the following offenses [translations]:
Vergewaltigung und sexuelle Nötigung [rape and sexual assault]
davon: [of which]
Vergewaltigung/sexuelle Nötigung überfallartig (Einzeltäter) [... suprise attack by individuals]
Vergewaltigung/sexuelle Nötigung überfallartig (durch Gruppen) [... suprise attack by groups]
Vergewaltigung/sexuelle Nötigung durch Gruppen [... by groups]
Sonstige Vergewaltigung [other rape]
Vergewaltigung/sexuelle Nötigung mit Todesfolge [... fatal]
Sonstige sexuelle Nötigung [other sexual assault]
Beleidigung auf sexueller Grundlage [insult on sexual basis]
Since these are 8 different tables, I don't want to pick one to quote here.
There is a related small inquiry of Ina Scharrenbach (CDU), a opposition politician in the parliament of North Rhine-Westphalia. Cologne is located in the state of North Rhine-Westphalia. The inquiry and it's response by the government of North Rhine-Westphalia provide more context for the statistics.
The inquiry makes the following remark about the statistics (page one, last paragraph):
It is also surprising that, without further explanation, the offense type "rape/sexual assault as surprise attack (by groups)" in December 2015 as in January 2016 shows no entry for the police department Cologne, whereas the number of cases known in June 2017 is indicated as "17".
Note: I presume that June 2017 should be June 2016.
This is where the number 17 comes from. It is the highest monthly number for this particular offense. It is true, that the statistics list 17 cases of this particular offense for June. However there are also 9 cases for for April. So the quote:
Just 17 cases went into the official police record - in April and June.
Is clearly false.
But furthermore, the inquiry requests clarification on why the April and June statistics are so high and how this is related to the Silvesternacht. The official answer is by the minister of interior, who is basically the head of the police. (Response to 4. on page 3):
For "security in focus", data on crime development are used. This is
based on the results of the police investigations at the time of the
final submission to the public prosecutor's office. At the time of the
complaint, the facts of the allegation of the offense, suspects and
victims are still unclear in many cases. For this reason, the tabular
assignment is basically made to the month in which the investigation
is handed over "to the judiciary". This is not always the same as the
month in which the offense was committed. Because of the large number
of criminal complaints and references to criminal offenses in the
Silvesternacht in Cologne as well as the related complex investigation
requirements, the increases in the number of sexual offenses reported
for the months of April and June 2016 for the district of the police
center in Cologne are exclusively due to the acts of the
The inquiry further requests clarification in which category, the 1150 criminal complaints are shown. The response (question 2 on page 2)
With regard to the statistical collection criteria, a detailed
illustration of the criminal charges in connection with the deeds of
the Cologne Silvesternacht is however not possible.
Now it is not clear whether all cases from the Silvesternacht went exclusively into April and June or also into other months. And it is not clear in what categories the cases went.
If we assume for a moment, that all cases went exclusively into April and June and those month had mean number of cases otherwise, there would be 284 cases from the Silvesternacht in the statistic. (back-of-the-envelope calculation) Again, this is only to get a rough feeling on the order of magnitude on how the Silvesternacht influenced the police statistics.
Is it true that the official crime statistics did not include the crimes from New Year's Eve sexual assaults?
No. It is included in the April/June statistics and they do show a significant increase. It was officially confirmed that this increase is due to the Silvesternacht.
Has this ever been explained/corrected?
There is an official explanation. However, there is no clear statement about the specific quantity and classification of those cases.
And does this imply that refugee and immigrant crimes are regularly falsified in official statistics?
No, it does not imply such thing. One might infer the opposite, but in any case it would be a stretch to derive a regular falsification or the lack thereof based on this singular issue.