Short answers, pulled from the story.
From the British perspective, the war began on the 18th of May 1803 when Britain declared war on France after the breakdown of the Peace of Amiens. The main phase of coalition fighting ended with the Treaty of Pressburg on the 26th of December 1805, though the subsequent French invasion of Naples continued into July 1806. Historians differ on whether the Naples campaign is part of the Third Coalition or not.
The Third Coalition comprised the United Kingdom, the Austrian Empire, the Russian Empire, Naples, Sicily, and Sweden. Sweden entered into an alliance with Britain in December 1804, Russia formally joined on the 11th of April 1805, Austria on the 9th of August, and Naples-Sicily on the 11th of September. Prussia remained neutral throughout.
On the 2nd of December 1805, Napoleon's forces defeated a combined Austro-Russian army east of Brünn in what became the decisive engagement of the war. Napoleon deliberately weakened his right flank to draw the Allied attack, then drove Soult's IV Corps through the Allied centre to seize the Pratzen Heights. The Allied army broke in a general panic, losing around 25,000 killed and wounded and 25,000 captured.
Signed on the 26th of December 1805, the Treaty of Pressburg required Austria to recognize French gains from the earlier treaties of Campo Formio and Lunéville, cede lands to Bavaria, Württemberg, and Baden, pay a war indemnity of 40 million francs, and surrender Venetia to the Kingdom of Italy. The Russian army was permitted to withdraw with its arms and equipment through hostile territory.
The Ulm Campaign ran from late September to the 20th of October 1805, when General Mack surrendered the bulk of the Austrian army encircled at Ulm. Napoleon feinted through the Black Forest while swinging 210,000 troops in a wide arc to trap Mack. The campaign ended with the Austrian force laying down its arms, with only around 10,000 troops escaping.
The Calabrian insurrection of 1806 was the first sustained guerrilla war the French encountered in the Napoleonic Wars. After French supply failures led General Reynier to seize food from starving peasants, a popular revolt spread across Calabria that was not suppressed until 1807. The French response, relying on terror tactics, foreshadowed the far larger guerrilla conflict Joseph Bonaparte would face in Spain during the Peninsular War.