How did the Germans make their way to Africa?

The main supply route of the Germans (and Italians) was by rail to ports in southern Italy, then by sea to Tripoli, or occasionally ports further east in Libya as they were occupied. The Italian merchant marine provided the bulk of the required shipping.

Didn't the British control the Mediterranean?

The British had strong forces at Gibraltar in the Western and Alexandria in the Eastern Mediterranean. However the Central Mediterranean was disputed, and at times dominated by Axis forces. German airpower and the Italian navy allowed them a measure of local superiority vs. the widely stretched Royal Navy. Malta was a crucial post in WWII because it gave the British an air and submarine base athwart Axis supply lines. When Malta was cut off, and nearly starved, the Axis were able to insure their supplies to Libya while preventing British convoys between the Eastern and Western Mediterranean.

In 1941, how did German soldiers consider a posting to the North Africa Campaign versus a posting to the Eastern Front?

In 1941 the Eastern Front had just begun (Barbarossa was in June) and achieved much early success. The eastern front did not yet achieve its reputation as a meatgrinder until the Stalingrad campaign the following year. The Desert on the other had, was not a very desirable posting. It was overseas, had poor sanitation and a good amount of disease (Rommel himself was one of many evacuated for Amoebic Dysentary), pitched battles, was considered a sideshow, and involved service with distrusted Italian troops.

You might try making your other questions their own posts.