Does Islam seek to establish caliphate to conquer the world?
The Arabic word khalifa (caliph) refers to a successor/vicegerent. Similar to the pope in Catholicism, the khalifa is responsible for continuing and leading the task of reformation and moral training that the prophet leaves behind after his demise, and for being the symbol of unity. Accordingly, Prophet Muhammad prophesied the advent of Khilafat after his passing:
Prophethood shall remain among you as long as Allah shall will. He will bring about its end and follow it with khilafat on the precepts of prophethood for as long as He shall will and then bring about its end. A tyrannical monarchy will then follow and will remain as long as Allah shall will and then come to an end. There will follow thereafter monarchical despotism to last as long as Allah shall will and come to an end upon His decree. There will then emerge khilafat on precept of prophethood. 
In the same prophecy, Prophet Muhammad laid down a timeline of future events: monarchy and tyrannical rule would follow khilafat, only for khilafat to be reinstituted in the Latter Days. Upon Prophet Muhammad’s demise in 632 CE, the “Rightly-Guided” khalifas–the first four khalifas succeeding Prophet Muhammad–assumed leadership of the Muslim Ummah, one after another. Criticism of some of the caliphs following the “Rightly-Guided” caliphate, while sometimes somewhat legitimate, has no bearing on Islam as Prophet Muhammad himself proactively disassociated himself from them in the prophecy above.
Fulfilling the above prophecy of the Prophet Muhammad, khilafat on the precept of prophethood was established in the Ahmadiyya Muslim Community in 1908 to continue the mission of reformation through love and amity started by Mirza Ghulam Ahmad (1835-1908), the Messiah of the latter days. The Community’s membership is in the tens of millions and spans 200 countries—all united under one khalifa.
Even when faced with bitter persecution, the Ahmadiyya Muslim Khilafat requires Ahmadi Muslims to be steadfast. Categorically rejecting all forms of religious violence, political supremacy, dictatorship, or authoritarian rule, the Ahmadiyya Muslim Khilafat champions the cause of the dispossessed through international humanitarian efforts, service, and dialogue.
 Musnad Ahmad bin Hanbal. Mishkat, Book: Riqaq, Chap. Al-Andhar Wa’l-Tahdhir.