OK, I had a chance to think out my proposals for tweaking the locations of Taoism, Islam, and Christianity. I took a turn when I started thinking out Christianity a little more, but I'll post the ideas and see what everyone thinks.
Taoism
The historical founding of Taoism is in the 4th century BCE, and schools are definitely active before 1 CE. This puts it into the Classical Era, not the Medieval. I suggest the following:
Taoism
Location: x34 y15
Requirements: Philosophy (+?)
This would put Taoism where Sikhism is being moved out.
Islam
The historical founding of Islam and its first great wave of expansion is before 750. I think this belongs more to the early Medieval Era, instead of the late Medieval where we have it now. Education is definitely a late Medieval tech, and a lot of the early spread of Islam was by conquest. I suggest this change:
Islam
Location: X36 Y7
Requirements: Feudalism AND Fundamentalism AND Resurrection
Christianity
This one is a little more complex, because I think there are three events that need to be included: the Christianizing of Rome, the East-West Schism, and the Protestant Reformation. This is one approach, and that is an explicitly Christian Reformation. I could see another scenario where any politically powerful, institutionalized religion undergoes a similar Reformation.
As it stands, one could not construct a late-Classical scenario with a Christian Roman Empire, because the Christianity technology would push Rome into the Medieval Era. I wouldn't count 3rd-century Judaea, either conquered or vassaled, as medieval.
So, there would be three technologies to cover the various forms of Christianity.
Christianity
Era: Classical
Location: X33 Y5
Requirements: Judaism AND Philosophy
Founds Christianity
Catholicism
Era: Medieval
Location: X38 Y1
Requirements: Christianity AND Papacy
Founds Catholicism
Protestantism
Era: Renaissance
Location: X44 Y1
Requirements: Catholicism AND Humanism AND Printing Press
Founds Protestantism
For the most part, the three strains of Christianity would act as differing religions, but they would share the same Holy City. (I'm not sure of all the details here). The biggest change would be that on the founding or first spreading of a new Christian denomination in your territory and one of the previous Christianities was your state religion, you could force a conversion; all of your cities would lose the previous Christianity and gain the new one. It's a way of implementing the
Cuius regio, eius religio idea.
Also, at the time of the discovery of Catholicism, the basic Christianity converts to
Orthodox in name only. None of its statistics change. So you can then stay Orthodox or convert to Catholicism. I imagine Catholic religious buildings being better at

(more attention to art) and Protestant religious buildings being better at

(closer to the common people).
Of course, if we radically redesign religion, then this may become moot, but I'm just following up on my previous suggestion.