Does Allah say Muslims and Islam will rule the world?

Yes, this is clearly and categorically promised in the Qur'an to the Muslim polity who holds the Islamic world view and follows the deen of Islam. In surah al-Nur, revealed in Medina around the time of the Battle of the Trench (Khandaq), Allah stated: وَعَدَ اللّٰهُ الَّذِيۡنَ اٰمَنُوۡا مِنۡكُمۡ وَ عَمِلُوا الصّٰلِحٰتِ لَـيَسۡتَخۡلِفَـنَّهُمۡ فِى الۡاَرۡضِ كَمَا اسۡتَخۡلَفَ الَّذِيۡنَ مِنۡ قَبۡلِهِمۡ وَلَيُمَكِّنَنَّ لَهُمۡ دِيۡنَهُمُ الَّذِى ارۡتَضٰى لَهُمۡ وَلَـيُبَدِّلَـنَّهُمۡ مِّنۡۢ بَعۡدِ خَوۡفِهِمۡ اَمۡنًا​ ؕ يَعۡبُدُوۡنَنِىۡ لَا يُشۡرِكُوۡنَ بِىۡ شَيۡـئًـا​ ؕ وَمَنۡ كَفَرَ بَعۡدَ ذٰ لِكَ فَاُولٰٓـئِكَ هُمُ الۡفٰسِقُوۡنَ "Allah has promised to those amongst you who believe and do righteous actions that He will certainly grant them khilafah (succession) on the earth, as He granted it to those before them; and that He will grant them authority to practice their religion, the one that He has chosen for them; and He will change their state from one of fear in which they lived to one of peace and security. They will worship Me alone, not associating any partner with Me. But whosoever disbelieves after this, then they are the rebellious ones." (Qur'an 24:55) The famous fifteenth century mufassir Ibn Kathir (d.774H) said about this verse: "This is a promise from Allah - the Most High - to His Messenger (saw) that He will make his Ummah the khulafa' (successors) upon the earth. Meaning that they will be the leaders and rulers over mankind, by which nations will be corrected and to whom mankind will submit. And that He will replace their state of fear in which they lived, to one of security and dominion. Allah - the Blessed, the Most High - has fulfilled His promise, and to Him belongs all praise and thanks. For the Prophet (saw) did not die until Allah made him conquer Makkah, Khaybar, al-Bahrain, the remaining lands of the Arabian peninsula and the whole of Yemen. He took jizyah from the Magians of Hajar and from those of the surrounding areas of Syria. The Prophet (saw) received gifts from Heraclius, the king of Rome; the ruler of Egypt and Alexandria, the Mawqawqis; the kings of Oman and Negus, the king of Ethiopia, who became king after Ashamah died - may Allah have mercy upon him and grant him nobility." This is supported by a number of ahadith on the issue, some examples are cited below: عَنْ تَمِيمٍ الدَّارِيِّ قَالَ قَالَ رَسُولُ اللَّهِ صَلَّى اللَّهُ عَلَيْهِ وَسَلَّمَ لَيَبْلُغَنَّ هَذَا الْأَمْرُ مَا بَلَغَ اللَّيْلُ وَالنَّهَارُ وَلَا يَتْرُكُ اللَّهُ بَيْتَ مَدَرٍ وَلَا وَبَرٍ إِلَّا أَدْخَلَهُ اللَّهُ هَذَا الدِّينَ بِعِزِّ عَزِيزٍ أَوْ بِذُلِّ ذَلِيلٍ عِزًّا يُعِزُّ اللَّهُ بِهِ الْإِسْلَامَ وَذُلًّا يُذِلُّ اللَّهُ بِهِ الْكُفْرَ Tamim al-Dari reported: The Messenger of Allah (saw) said, “This matter will certainly reach every place touched by the night and day. Allah will not leave a house or residence but that Allah will cause this deen to enter it, by which the honorable will be honored and the disgraceful will be disgraced. Allah will honor the honorable with Islam and he will disgrace the disgraceful with unbelief.” (Ahmad 16957, Tabarani 1280, Hakim 8326) Thawban narrated the Messenger of Allah (saw) said: "Indeed Allah gathered up the earth for me, so that I saw its east and west. Indeed, the dominion of my Ummah will reach what gathered up for me, from it." (Sahih al-Muslim 2889, Abu Dawud 4252)