1. What caused the decline of the Ottomans beginning with the 17th century up to the beginning of the twentieth century?
The reign of Mahmud II directly led to the decline of the Ottoman Empire, as his decisions hung over the empire for generations. First and foremost, Mahmud II abolished the Janissary corps in 1826 in order to reform the Ottoman Empire's army and modernize the nation of Turkey, which also led to increasing Western European influence in the nation. Nationalism began to rise even more, often centered around religion; following the Tanzimat reforms of 1839, the self-contained millet courts were removed in favor of a nationwide system.
The terrible economy of the Ottoman Empire at the time also contributed to its decline. Taxation was high, and this left the lower classes with a substantial amount of social oppression. Many of these oppressive charges and taxes were changed under Mahmud's reign; though that is ostensibly a good thing, these varying changes which all occurred right after the other substantially disrupted existing systems and affected the economy. Without a functioning economy that adjusted to changes in financing like the British and French could, the Ottoman Empire continually shrunk until Russia and other nations continually chipped away at their borders.
2. How unified, or divided, was the world of Islam between, roughly, 1600-1875. Be sure to explain the reasons for your conclusions.
I would say that the world of Islam, especially as it relates to the Ottoman Empire, was fairly divided from 1600-1875. Many different countries, and even communities, had their own idea of Islam that they would exercise throughout their own domain. The millet courts were legal courts in which communities were permitted to exercise self-rule under one single system. This system was maintained largely through this period of time for Islam, permitting each community to have its own notion of diverse beliefs that comprised their religion. In this way, Islam practiced a significant degree of religious pluralism throughout the Ottoman Empire.
Despite these differences, however, there was a substantial amount of unification; due to the Sultan of the Ottoman Empire doubling as the Caliph, Ottoman law did not differentiate between ethnicities or citizenships, allowing all Muslims, regardless of sect, to be protected under the same law. This is one of the factors that made the Ottoman Empire a great haven for Muslims. However, with the rise of Arab nationalism in 1821, secular policies from the Young Turks and divisive political decisions started to split apart various Muslim sects, dividing them substantially once again. While other sects of Muslims did unite against the Arabs, the resulting infighting caused significant damage to Mecca, and led to the eventual dismantling of the Ottoman Empire as a Pan-Islamic state.