Are the rumors true that Recep Tayyip Erdoğan is supporting ISIS and if they are then why would he betray the international coalition against the Islamic State?

Answer 1

There are two questions here:

Is it true that Erdogan is supporting the Islamic State?
Definitive information about secretive deals is nearly impossible to come by, so, with the exception of people in the Turkish and Islamic State governments, nobody knows with certainty about what deals may be structured between those two groups. However, most political commentators and journalists in the region do believe that there is sufficient evidence to establish a link of at least non-aggression and possible monetary support. The clearest evidence of a functional relationship between Turkey and Islamic State are the repeated sales of petroleum from Islamic State to Turkey and Turkey's non-aggression against Islamic State, even when the latter has crossed the Turkish border on certain occasions.

Why would Erdogan support the Islamic State, especially when this violates NATO and Anti-IS Obligations?
Quite simply, Erdogan sees Assad as the greater evil and will do anything (including supporting Islamic State to achieve this goal). Erdogan has delusions of grandeur and a desire to resuscitate the Ottoman Empire. In this vein he believes that he might be able to exert some kind of cultural or political control over the Islamic State because of their common bonds as pious Muslims that he could not exert over Assad, since Assad was an Arab Nationalist and Non-Muslim. (Assad is an Alawite.)

As for Erdogan's "obligations" in the Coalition Against Islamic State and Turkey's membership in NATO, Erdogan is wagering that neither the US nor European countries will push Turkey to actually abide with its treaty obligations. He believes, correctly given recent politics, that since the US needs Turkish bases like Incirlik to launch its aerial assault on Islamic State and that European countries need Turkish cooperation to halt the flow of Muslim migrants and refugees, these countries will not press their luck.