Why women cover their heads for religious prayers? Why contradictory beliefs in North and South?

Good question...though I don't have an appropriate answer to it. For what I know is that in Northern part of India women are asked to cover their heads only when they are married or have kids. Unmarried girls as far as I know are not supposed to cover their head. But I really don't know the why part..part of tradition i guess, which nobody chose to explain and nobody answered but has been handed down for generations to come.

Half of the answer lies in your question itself. Like Kerala and Tamilnadu, The practice of covering of the head is not there in Andhra and Karnataka. Yes, there was this custom of head covering in different sects of Christianity. But the modern christians are gradually dropping this practice during church prayers.
Then the answer for your " Why" is, all the customs and religious practices are created by humans.They are based on a particular time and age. And they are also not binding.
Religion varies from place to place. Even Ramayana and Mahabharat varies from place to place. With the time many additions and deletions were made by the self styled Gurus.
In Satya Harishchandra, the name of his wife varies from place to place. To my knowledge, it was Chandramati. But in North Indian Mahabharata, it is Taramati. This is an example to emphasize that religion and religious practices are man made and they mostly depended on the whims and fancies of those the then Gurus.
So this is the reason for variations in religious practices in the country.

All practices including religious change with time and also geographical distance. This is true about verything- food, clothes, worship, traditions. Why this happens is simply because humans are not factory products that conform to preset design.

I cannot find logic in religious beliefs and practices as desired by Sandhya. Religion is most illogical and baseless. This is worst than opium. So it suffices to say that it is natural that people will develop different food habits, dress, perceptions in different regions.

Well, even same religious people do praying and related things differently. Different people have their own belief and their own style.

The traditions being followed for a long time, kind of turns into beliefs.

I dont know about that.