Were there treaty clauses that gave these powers a way out?

The Franco-Russian Entente and the Triple Alliance were all defensive in nature. There was no obligation to enter hostilities on the side of a signatory if that signatory was the aggressor. Hence Italy's actions in July 1914; as Austria-Hungary and Germany were the aggressors, Italy was not obliged to support them.

Would Germany have gone to war if Russia didn't mobilise and/or Serbia initially reject the ultimatum?

For one thing, the Serbs did not reject the ultimatum. While they did not accept all of the points unconditionally, they did accept most them, the main modification being that a neutral party (specifically the Hague) should undertake the investigation into the assassination, not the Austro-Hungarian gendarmerie. In attacking the Serbs regardless of this, the Austro-Hungarians were the aggressors, and as the Triple Alliance did not cover Serbia, Germany and Italy were under no obligation to support the Austro-Hungarians. The Russians mobilized, but did not issue any declarations of war, with mobilization having only progressed a few days when the Germans issued their ultimatum and then declared war. Germany was under no obligation to go to war, but instead acted outside the bounds of the Alliance.

Would France have gone to war if Germany didn't declare war on France and attack first?

The Dual Entente, like the Triple Alliance, was a defensive agreement. However, this question is complicated by the fact that the Germans had no war plan or mobilization plan for a single or two front war focusing on Russia, as of 1913. After that year, if there was a war with France alone or with both Entente members, Germany's focus from the start would be the defeat of France. Technically due to the German declaration of war on Russia, France was obligated to support it's ally, but again, German war plans had already decided on "France first."

Would Britain have gone to war if Belgium was not attacked?

Britain did not have an Alliance with Belgium. The Treaty of London, signed in 1839 and reaffirmed in 1867 and 1871, obligated all signatories to defend Belgian sovereignty in the event that Belgium was attacked. So, had Belgium not been attacked, or even had the Germans greatly limited the scope of their Invasion, the British would have had no obligation to intervene. Moreover, the Cabinet was against intervention unless to secure Belgian sovereignty, and the House of Commons and public opinion were divided but leaning towards non-intervention.