Short answers, pulled from the story.
The French forces captured Cosseria Castle after a costly siege that lasted from the evening of the 13th of April until early the next morning. Provera capitulated because his men were out of food, ammunition, and water.
The French lost 700 men in their fruitless attacks on the 13th of April 1796. Specific deaths included Adjutant General Jean Quenin, General of Brigade Pierre Banel, and Sardinian commander Del Carretto.
Napoleon likely crafted the misleading report to cover up serious French casualties and how close he had been to having his plans seriously compromised. He later admitted to Colonel Joseph Costa that the siege was a mistake due to his impatience.
Giovanni di Provera led the weak Austrian Auxiliary Corps east of Millesimo on the 13th of April 1796. His forces numbered only about a thousand Austrians and Piedmontese under Provera and Del Carretto.
The surrender of the castle allowed the French offensive to continue toward Dego on the 14th of April 1796. This victory enabled the continuation of the French campaign across Liguria and shifted the strategic situation decisively in favor of France.