A group of Serbian officers, the Black Hand, did provide limited weapons and training to Mladna Bosna, but it seems the plot to kill the Archduke was very much the latter group's initiative. We know that the Serbian Prime Minister, Nikola Pasic, was aware of these cross-border operations and was seeking to put a stop to them, and that some sort of vague warning was given to the Austrian government prior to the assassination. Therefore, No, I would say that Serbian responsibility for the outbreak of WWI was at most quite limited.

July Crisis by Thomas G. Otte The Trigger by Tim Butcher

Yes and no. In the early 1900s, a secret society known as the Black Hand composed of Serbian military and intelligence officers was formed to promote pan-Serbianism with the goal to liberate Serbs under the control of neighboring powers, such as Austro-Hungary, which occupied Bosnia in 1908. The goal of uniting Serb-inhabited territories was implemented through recruiting, training and supplying guerrilla fighters and saboteurs; among these men was Gavrilo Princip, the man who shot Archduke Ferdinand. Presently, it is understood by historians that only a portion of the Black Hand planned and supported the operation to kill Archduke Ferdinand. These officers knew that the action would spark a war with the Austro-Hungarians but they hoped that Russia, the self-proclaimed guardian of slavic peoples, would come to their aid. They were envisioning a continuation of the 1912-1913 Balkans Wars, which was a regional conflict that forced the Ottomans out of Europe. In this case, they sought to force the Austro-Hungarians out of Bosnia. Whether they could have foreseen the response of Germany to the situation was unlikely.

Sources: Hew Strachen, The First World War (Viking Adult; 1st edition, 2004).

"Balkans Wars," from International Encyclopedia of the First World War. http://encyclopedia.1914-1918-online.net/article/balkan_wars_1912-1913

It depends on what you mean by "their fault." If you take the fault for the conflict to be that the Great Powers at the time all had competing concerns and goals, and all that it took for them to go to war was a precipitating cause, then Serbia isn't at fault. Under that rubric, if it wasn't the assassination of Franz Ferdinand, then it would have been some other crisis that caused a general mobilization and war.

However, if you believe that only a specific type of cause - i.e. one that allow for each of the Great Powers to justify their intervention - that could lead to an outbreak of World War I, then Serbia bears at least a portion of the blame. Serbia was heavily interested in expanding it's territorial range in the Balkans, and the Serbian government looked for ways to cause that to happen, Serbia's participation in the Balkan Wars being the most prominent example. There is evidence that the Serbian government, or at least certain elements of it, contributed heavily to the assassination, and that the attempt was done with full knowledge of those individuals in official government positions.

I would highly recommend reading "The Sleepwalkers: How Europe Went to War in 1914" by Christopher Clark. It is extremely informative, and will provide quite a bit of insight into the usual Anglo-Franco-American view that the Central Powers were the sole instigators of the war. In particular, Austria-Hungary comes out looking like the least culpable power out of all of them, with the French contrastingly looking pretty poor. I think you'll find that the book provides a lot more nuance as to whose "fault" it was than the standard story.