The Muslim world was essentially entirely under a unitary imperial state until the Abbasid Revolution (747-750 CE) in which the Abbasids (750-1258) overthrew the Umayyads (661-750). The Abbasids inherited most of the Umayyad territories with the exception of al-Andalus (Islamic Iberia) and Morocco. In the decades after the Revolution, the Abbasids consolidated and strengthened their rule. However, by the mid-late-9th century the power of the Abbasids was in palpable decline, but they nonetheless retained their important status as Sunni caliphs until the destruction of Baghdad by the Mongols (1258). The dynasty continued to exist as titular caliphs in Cairo until the Ottoman conquest of the Mamluk Sultanate in 1516-17, from which the title of caliph passed to the Ottoman dynasty.

To fully answer your question in detail would require whole books because the relationship between the Abbasid Caliph and other Muslim states varied widely depending on religious, political and geographical factors. Furthermore, the exact degree of independence from Baghdad, theoretical and in actuality, also varied widely. The relationship could be hostile, as was the case with the rival Shi'i caliphate of the Fatimids (909-1171). It could also be relatively amicable, as was the case with the Samanids (819-1005) who ruled the eastern Iranian lands into Transoxiana and whose rule was sanctioned and legitimized by the Abbasids. There were instances in which the Abbasids were effectively reduced to vassaldom by another dynasty, as was the case with the Shi'i Buyids (945-1055). The relationship of the Ottomans with other Muslim states is an entirely different topic.

Now, regarding the relationship between the Abbasids and Seljuks: The Seljuks along with other Oghuz Turkic tribes migrated into southern Central Asia (Transoxiana) and, later, into the Middle East beginning in the mid-10th century. The decline of the Samanid state facilitated these migrations. The Muslim rulers of Khwarezm (the delta region where the Amu Darya enters the Aral Sea) annually organized campaigns into the Eurasian steppe called faghburiyya which were effectively "counter-raids" and took back booty, particularly slaves. The Oghuz Turks of the 10th century, like other nomadic pastoralists, partook in raiding in pillaging settled folk and there is no shortage of stories of their uncouthness and barbarism. It should also be mentioned that the state religion of the Samanids was Sunni Islam of the Hanafi madhhab, which concurrently became the dominant form of Islam among the Turks - this remains the case to this day.

The grandson of the eponymous Seljuk ibn Duqaq, Sultan Toghril Beg, captured Baghdad from the Buyids in 1055 and his rule was later legitimized by the Abbasids. The Seljuks framed themselves as the defenders of Sunni Islam against its enemies. In the years, decades, and century before the Seljuks captured Baghdad, Shi'ism seemed politically ascendant. The Sunni caliphs were subjects of the Twelver Shi'i Buyids. The Isma'ili Fatimids had an empire that stretch from Syria to the Atlantic and included Sicily and the Hijaz at its apogee. In eastern Arabia there were the Qarmatians, who raided Mecca in 930, killing pilgrims. Isma'ili missionaries spread their message in numerous corners of the Muslim world, particularly in regions such as northern Iran.

There was intermarriage between the Abbasid dynasty and the Seljuks. In 1058, the daughter of the caliph al-Qa'im married Toghril Beg. Sultan Malikshah's daughter and Caliph al-Muqtadi married in 1087. The two dynasties also shared in occasionally getting murdered by Isma'ili Assassins. Examples of those who met this fate include Caliphs al-Mustarshid (d. 1135) and al-Rashid (d. 1138). The Seljuk vizier Nizam al-Mulk (d. 1092) also was killed by assassins. Of course, Seljuk Sultans and other Sunni rulers were not above hiring Assassins for, well, assassinating their political enemies.

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Sources:

A. C. S. Peacock, The Great Seljuk Empire

Christian Lange and Songul Mecit, The Seljuqs: Politics, Society and Culture

Josef Meri, Medieval Islamic Civilization