* I really think the first sentence needs to be more comprehensive in terms of defining the subject. It is a very short sentence and doesn't really provide much information. What about something like "Köln was a German light cruiser that was operated between 1929 and March 1945, including service in World War II. (was she the lead vessel in her class? I thought that was Königsberg?) She was operated by two German navies, ..."
I have a query about the format of the article title. According to WP:NCSHIP Karlsruhe should be italicised in the title.
Also, the DKM in the infobox is not consistent with her sister ships (one doesn't have a prefix there, one has KMS. It was my understanding that Kriegsmarine ships didn't have a formal prefix, but that some authors use them. Perhaps it would be better to be consistent with the article title and drop the prefix? OK.
2a. it contains a list of all references (sources of information), presented in accordance with the layout style guideline.
OK
2b. reliable sources are cited inline. All content that could reasonably be challenged, except for plot summaries and that which summarizes cited content elsewhere in the article, must be cited no later than the end of the paragraph (or line if the content is not in prose).