Which caliphate are we talking about ('Abbasid, Fatimid...?) Any time period in particular?

By the time of the First Crusade the Abbasid caliphs had mostly been figureheads for some time.

While slave soldiers had been introduced earlier, starting with the Anarchy at Samarra the Caliphs became utterly dependent upon that relationship with their soldiers, including for their own lives. While after the anarchy there was some recovery of power, by the end of the 9th century the Abbasid territory was characterized by the fracturing of independent local dynasties, while at home the Turkic military Amirs acted as regents with total control over the Caliphate.

This loss of power was most clearly established in fact when the Buyid's, a dynasty of Shia Turkic warlords, conquered Baghdad in the 10th century.

From that point on, the Caliph's personal power was almost exclusively reserved to the religious sphere, with the Buyids and then the Seljuks preserving the Caliph's title and position for that purpose.

The issue of taxation is an interesting one, because in an Islamic context in this period there were actually quite a lot of restrictions placed on the ability to tax and still be within the bounds of the Shariah. The most common form of taxation was the Iqta, which is often compared with European Feudalism (which itself is of course hotly debated as to what it was and whether it existed but that's a bit beside the point). Without getting into too much detail on the various forms the Iqta could take, it was basically a system of tax farming where the ruler or prince or general or family member or high-level bureaucrat, would directly or indirectly be able to collect the taxes on a particular tract of land.

By the time of the first crusade the ability to dole out these Iqtas was held exclusively by the Turkic Amirs.

All of this isn't to say that the Caliph wasn't still of great symbolic importance. The oaths between the Amirs and the Caliph were a hugely important part of the legitimacy of the state, and the Caliphs name and titles were still stated every Friday in every mosque during prayers. When Saladin deposed the Fatimid Caliphs in Egypt returning the Caliphs name to Friday prayers was one of his first acts.