Lubeck was a single city state that was the center of the Hanseatic League. During their high times (nearly 100 years prior to this conflict), most trade from the baltic sea flowed through Lubeck or on ships made by Lubeck. They sacked Copenhagen and a few Danish cities. Simple theory, control the trade = control the money = control the power. At one point in time, any boat in the Baltic waters was likely made or designed in Lubeck. However the 1500s saw economic crisis's and increased competition from further west (House of Habsburg) saw their power wane.
The counts war was actually initiated by Lubeck and can be considered a war of succession after Frederick I (king of Norway and Denmark) passed. Fredrick had co-ruled with Christian II and upon Fredrick I's death, Lubeck was one of the supporters of returning Christian II to the throne. Christian III however would become the favored of the Jutland nobles...though the support is mostly along religious lines (Christian II gaining his support from the Catholics while Christian III drew support from Lutherans). Lubeck invaded Holstein (I believe to free Christian II who was held there), triggering the counts war.
Christopher, Count of Oldenburg (https://en.wikipedia.org/wiki/Christopher,_Count_of_Oldenburg) is where the name 'counts war' comes from, was a major supporter of Christian II and was hired by Lubeck to bring Christian II back to the throne. The Count ordered Skipper Clement (https://en.wikipedia.org/wiki/Skipper_Clement) to initiate a revolt of the peasants of Jutland vs the nobility there that supported Christian III. This actually worked in the beginning and the peasant army burnt many nobles houses and successfully routed the Nobles army there.
I'm trying to find more details on it but am unable to come up with anything better than 'Christian III forced peace with Lubeck' (how that came about I'm still unsure, "shrewd negotiation" is another term I find there, appears to be a diplomatic peace with Lubeck and the Hanseatic league). With Lubeck pacified, Christian III was able to free up his soldiers and general to suppress the peasants revolt in Jutland (the peasants were poorly organized and even poorer equipped, the professional force Christian III commanded had little issue suppressing the rebels from there).
Hard to say Lubeck won or lost in this conflict. To your questions:
I can't find maps that shows all the territories Lübeck control, such as the Island of Bornholm was under Lübeck's control in the war. Is there information on available on this?
Look up Hanseatic league. Technically Lubeck didn't control these other territories and were part of the league.