Short answers, pulled from the story.
Operation Sonnenblume, also known as Operation Sunflower, was the dispatch of German and Italian troops to North Africa in February 1941. It followed the destruction of the Italian 10th Army by British and Commonwealth forces during Operation Compass, and was ordered by Hitler via Directive 22 on the 11th of January 1941 to send a blocking detachment to stabilise the Axis position in Libya.
Generalleutnant Erwin Rommel commanded the Deutsches Afrikakorps during Operation Sonnenblume. He arrived in Libya on the 12th of February 1941 and was formally ordered to defend Tripoli and Tripolitania, albeit using aggressive tactics.
The first units of the Deutsches Afrikakorps arrived at the Libyan port of Tripoli on the 14th of February 1941. The first German tanks arrived on the 20th of February, and Panzer Regiment 5 completed its arrival in two convoys from the 8th to the 10th of March 1941.
The British position in Cyrenaica was critically weakened before Sonnenblume began. The most experienced British units had been transferred to Greece in Operation Lustre, the 3rd Armoured Brigade had tanks too worn out to fight effectively, and British commanders underestimated Rommel's willingness to attack before expected. Wavell later wrote that he had not budgeted for Rommel after his experience fighting the Italians.
On the 8th of April 1941, Major-General Michael Gambier-Parry and between 2,700 and 3,000 British, Indian, and Australian troops surrendered to Major-General Pietro Zaglio of the Italian Pavia Division at Mechili. Lieutenant-Generals Philip Neame and Richard O'Connor were also captured separately that night after taking a wrong turn near Martuba.
No. Tobruk held against Axis attacks. Rommel invested the port on the 11th of April 1941, but attacks on the 13th-14th and 16th-the 17th of April were repulsed by a garrison of about 25,000 British and Australian troops. On the 2nd of May, Rommel accepted that the Axis force was insufficient to take Tobruk.