Okay, so this deserves an answer. Since I answered something similar in the past, I'll try to answer this as well. I don't have really extensive knowledge on this, but I will try my best.

A) So, I'll start at the end of your question. Germany is expected to follow through on this because its new government signed the treaty. When Wilhelm abdicated and fled, he more or less forfeited a lot of personal effects, assets and land. The prussian government seized basically all of the Hohenzollern's property in Germany. They had to pay them back for it, in 1926. See: link This was all legitimate as it happened to the Habsburger (i refuse to use the anglicised name) in Austria as well. Everything the "crown" owns is property of the state anyway. So that's how they could expect the german government to make good on this deal.

B) Why would they give Wilhelm II an invalubale copy of an early Koran? The Ottoman government was in a massive crisis between 1908 up to 1913 because of a coup on the inside of the so called "Young Turks" and the permanently smoldering powder keg of the Balkans. So they needed support. Germany didn't have much in terms of allies besides Austria, so they were more than happy to help. Economically, german industry was heavily invested in the Ottoman provinces. They build railroads (Bagdadbahn und Hedschasbahn - roughly translates into Baghdad railroad and Hedjaz railroad) and invested in other parts as well. Besides being a prestige project, especially the Bagdadbahn was meant as a landbound challenge to the british domination of the seas. If you have any doubt about how important the project was, just think of it as a railbound Suez canal with access to the oilfields of the region and you'll understand the massive importance for both sides. For the Ottoman empire this kind of investment and prestige was paramount. The empire was falling apart and it needed success, especially economically. The Ottoman empire was bankrupt in 1875, so this kind of investment, especially longterm, meant a lot. To ensure good relations, Emperor Wilhelm II. visited Istanbul three times. Once in 1889, once in 1898 and finally in 1917. Wilhelm was a big fan of the Ottoman Empire as opposed to Bismarck and wholeheartedly supported them. He is supposed to have received the Koran during the visit of Mehmed V to Berlin in 1917. But sources are rare, I couldn't find any primary sources so far.

C) Okay, now for the tricky part. For reference, every copy of a Quran of Uthman in the whole world and their provenance is a matter of public record. Link - That's how important they are. I found one mention of this particular one in "Der Krieg nach dem Krieg: Deutschland zwischen Revolution und Versailles 1918/19 von Andreas Platthaus". But, there seem to be two Tashkent-Manuscripts, and there is little on how they got to Berlin and back.

D) Final thoughts and some speculation: I think it is far more likely that Wilhelm II was presented with one of the 50 facsimiles of the St. Petersburg Manuscript which resides now in Tashkent. However, seeing as how important the Hedjaz was for the british plans for Arabia, this was probably meant as a gesture of good will towards the Sharif of Hedjaz. The politics here are pretty well known, just watch Lawrence of Arabia.

EDIT: Thanks for the question. This was really fun and interesting to research. EDIT 2: Fixed an omission.