Yeah they are controversial, I'm not saying there's a default position on these issues. I'm saying either side has to justify their stance.
The PSR is that everything has an explanation or reason for existing, which is different to the first cause argument. For example, the PSR can be problematic for theists, because they have to address the issue of an explanation for God. Some say he's explained by himself, some say he's explained by his essence, and some say there is no explanation (brute fact).
Brute contingency is basically saying that contingent beings/properties can exist as brute facts (without any explanation).
Believing in brute facts usually goes hand in hand with rejecting the PSR. If I held these positions I would be an atheist.
As for why people believe the first cause must be God, or have deity-like properties, there's a whole range of arguments and it takes too long to explain. But good theists aren't simply saying 'a God is necessary, you need to prove otherwise' they usually at least justify believing in philosophical or deistic properties of God, but rarely justify belief in theological properties (such as the Trinity or that he loves us all).
The three standard arguments for God in contemporary philosophy of religion are the cosmological argument (probably the biggest one) the ontological argument (not many people use this) and the argument from design.
However, in my personal opinion (I stress that this isn't common knowledge or anything of the sort) I find these arguments insufficient on their own because like most atheistic models, they have to assume a lot of metaphysical propositions.