I have nothing to offer on what ships Essen had with him on the 9th of August, though it would be safe to assume that the entirety of his command was with him.
I do, however, recall reading from someone else on this site (wiking, IIRC) that the bulk of the Swedish Navy was on exercise in the waters off Stockholm, so Essen will likely come up against the shore batteries and fishermen of Visby.
All told, Essen will likely fail in his mission of preemptively destroying the Swedish Navy while at the same time bringing it into the war on the CP side in what will undoubtedly be a massive PR victory for the Central Powers.
The Swedish and German navies are able to sweep the Russians from the Baltic, giving the Swedish Navy free rein to support land operations in Finland.
The Russians are forced to divert forces from other fronts (presumably either the Austrian or Caucasian ones) in order to deploy suitable defense forces in Finland, hurting Russian efforts in whichever front they choose to withdraw forces from.
Russian forces in Finland suffer from logistics problems due to Finland's poor infrastructure. Swedish forces, however, operate with the logistical support of the Swedish and German navies, which should at least give them an edge in the coastal areas of Finland.
The Swedes are able to win the Finnish campaign and find themselves at the gates of either Viborg or St. Petersburg at the same time that German troops are marching through Narva.
Presumably, the presence of Sweden in the Central Powers helps German leadership to realize that their alliance is one which has the majority of its guns pointed east, not west. This leads to Moltke's replacement being a Russia-first general like Hindenburg rather than a France-first one like OTL's Falkenhayn. The Germans and the CPs in general devote more resources to the east, with the result being that the Gorlice-Tarnów offensive is intended from the very beginning to inflict massive defeats on the Russians, rather than being initially conceived as a pressure-relieving offensive as in OTL.
ATL's Gorlice-Tarnów offensive would also probably be conducted in concert with a Swedish offensive in Finland, an Austro-Hungarian one in Galicia (presumably the Austro-Hungarian situation is much better due to having less opposition in the 1914 Galician campaign), and an Ottoman offensive in the Caucasus. The Great Retreat turns into the Great Rout and the Russians are knocked out of the war about a year earlier.
Hindenburg being in the driver's seat means that the Verdun initiative never happens, so the German situation on the Western front is better. CP successes on the Eastern Front mean that Romania is in no position to join the Entente, while Serbia would probably fall by OTL.
I don't know what direct effect Swedish involvement would have on German USW policy, but Swedish-Americans in particular and Scandinavian-Americans in general would be more opposed to the war, presenting an additional domestic barrier to US entry into the war. It would also be much harder for the Entente to paint the CPs as the "bad guys", because the Russian violation of Swedish neutrality would either blunt or neutralize the negative PR effects of the German violation of Belgian neutrality.
All told, I believe that the Central Powers would have won WW1 if Essen had followed through with his raid.