Although I cannot attest to why the Ottoman Empire did not continue to progress along with various European States, I can answer your second question on what stopped, or rather slowed, Mediterranean Islam's scientific discoveries.

When people think of the "Golden Age" of Islamic advancement most of them are thinking of the period of rule by the Abbasid Caliphate before the Mongol destruction of Baghdad in 1258. The Abbasid Caliphate rose in 750 after the rebellion against the Ummayad Caliphate (for various reasons) and moved the capital from Damascus to Baghdad. Baghdad was located along a much more important route from the East to the West than Damascus was and the passage of information and learning through the city was capitalized upon by the Abbasid Caliphs. When the Caliph al-Mansur founded Baghdad, he also founded Bayt al-Hikma, or the House of Wisdom, meaning to compile any and all information he could. Bayt al-Hikma became the seat of this Islamic scientific advancement. Under other Caliphs such as ar-Rashid the Islamic Empire started to translate books from Greek, Latin, Sanskrit, Chinese, and Persian into Arabic. The Caliphs also started to invite delegations from other faiths and other empires to work under them. Knowledge was prized so much that great works became important spoils for war, and by the second half of the 800s, Bayt al-Hikma had become the largest library of books in the world.

So why did this end? Many peg the end of the Islamic Golden age in 1258 with the Mongol destruction of Baghdad. You know how in many books the author will highlight the destruction of a town by saying the rivers ran red with blood? Well survivors of Baghdad actually claimed the Tigris ran black with ink because of the amount of books tossed into the river by the invaders. Because of this one sack, the Islamic world lost their largest repository of information. Steven Dutch, author of The Mongols, wrote this regarding the sack:

Iraq in 1258 was very different from present day Iraq. Its agriculture was supported by a canal network thousands of years old. Baghdad was one of the most brilliant intellectual centers in the world. The Mongol destruction of Baghdad was a psychological blow from which Islam never recovered. Already Islam was turning inward, becoming more suspicious of conflicts between faith and reason and more conservative. With the sack of Baghdad, the intellectual flowering of Islam was snuffed out. Imagining the Athens of Pericles and Aristotle obliterated by a nuclear weapon begins to suggest the enormity of the blow. The Mongols filled in the irrigation canals and left Iraq too depopulated to restore them.

This utterly ended the height of Abbasid power, and although other Caliphates or Sultanates reined for periods of time (The Fatimid, The Ottomans, The Mamluks Sultanate), the Islamic world was never again as united. This partially explains why there was not another big "explosion" of knowledge, as few other rulers could provide the stability needed for such massive undertakings or control such vast and different swathes of territory and cultures.

Secondly I would like to address what you said with Islam starting to be "what it's today". (Mods, I am going to talk about the history of Wahhabism, but might briefly refer to current events. Let me know if I should remove those references) I am going to assume you are referring to events such as ISIS, Boko Haram, or various other "Islamic" groups decrying learning and "Western" knowledge. What is important to know about these groups is that they are extremists of the Wahhabi Sect, which you can read a little more about here (bear in mind the author is extremely anti-Wahhabi). Although I am not going to go into too much detail about it now, as I have to unfortunately do work, the Wahhabis follow a literalistic interpretation of the Qur'an, often violently acting against those who they view as acting in the wrong, such as the destruction of the Tomb of Hussein bin 'Ali in 1801, which was a holy site for Shi'ite Pilgrims and thus an "idol" in the views of the Wahhabis. The Wahhabis are a relatively new branch of Islam, originating only in the mid 18th century. However, it has grown to become the state religion of Saudi Arabia due to the early alliance between the House of Saud and Ibn Abdul-Wahhab, the founder of Wahhabism. Wahhabism gives credibility to the Saudi Regime, and in exchange the House of Saud promotes Wahhabism and "exports" it across the Muslim world. It is important to note that this sect of Islam is not representative of all Muslims, and other Muslim leaders, particularly Shi'ite or Sufi, have condemned Wahhabism multiple times since its founding through today.

Islam of today is still definitely contributing to Science. This is edging to close to modern history now, against the 20 year rule, but I would urge you to look into advancements made by Iran in the fields of medicine (particularly stem cell research) and aerospace. This Report talks of how scientific advancement in Turkey and Iran is at nearly 4-times the global average.

Good Readings to you, and I hope someone else can answer the first part of your questions!

Sources in addition to the papers I linked to:

A History of Islamic Societies by Ira Lapidus is a good book that I myself own. I would definitely recommend it to anyone looking for a good overarching view of Islamic progression.

The House of Wisdom: How Arabic Science Saved Ancient Knowledge and Gave us the Renaissance by Jim Al-Khalili. I unfortunately don't own this book myself, but have read small bits of it and it goes into more detail about the "Golden Age" of Islam.

Please feel free to ask questions or give me feedback! I am still relatively new to posting on AskHistorians and want to make sure I uphold their standards.