Germany had the largest army and the second largest navy in Europe but had very limited means to project power beyond its borders as it had not engaged in the rampant colonialism of other European nations such as France and Britain. As such, Germany under Kaiser Wilhelm II was desperate to strengthen and expand its borders in the face of French & Russian military buildup.

(French/Russians would of course say that their military buildup was in response to Germany's own increase in armaments)

While you are correct that relations between Germany & Austria were not great, Bismark played on mutual fears of Russian influence in the Balkans to manoeuvre both nations into signing the Dual Alliance in 1879. The Dual Alliance was a mutual defence pact in case of Russian aggression.

In 1882, the Dual Alliance was secretly expanded to become the Triple Alliance. This move was also driven largely by Bismark's desire to have Germany seen as a strong power, a force for peace and to avoid a war on two fronts. The Triple Alliance bound Italy to remain neutral in the event of a war with Russia and to assist if France attacked.

Without going too much further, when Austro-Hungary chose to take an issue with Serbian separatist movements via their ultimatum over Archduke Ferdinand's assassination, Russia (who had long been courting Serbia as a method to gaining influence in the region mobilised). Russian mobilisation triggered Austro-Hungarian mobilisation and invocation of the Triple Alliance. Germany mobilised and proceeded with its war plans.

(In those days, mobilisation meant troop movement and was very much seen as a highly aggressive action. War plans simply couldn't be changed once implemented and, for Germany, mobilisation meant attacking France, who was an ally of Russia, via Belgium).

The Ottoman Empire's army was in a shambles as you correctly point out. They signed the Turco-German Agreement in 1914 and the benefits to Germany were clear.

Firstly, the Ottoman Empire would not join on the side of the allies. Secondly, Germany would have access to Mediterranean sea ports. Finally, Germany could use the alliance to pressure Romania & Bulgaria to join the war on its side.

For the Ottoman Empire, they saw stunning German victories early in the war and assumed that there would be a short, victorious conclusion. They would also benefit from being allied to the strongest army in Europe.

To the best of my knowledge, the Austro-Hungarian Empire was not consulted over the alliance with the Ottomans.