The Evidence Required Is?

[Welsh cake]

But what all religions have in common is a belief in forms of reality other than the one governed by the laws of physics.

This does not alleviate the burden of proof in anyway, it actually adds to the baggage because now you've got two huge claims "a god exists" and "alternate realities exist". Lumping them both together does not make the task of fulfilling your obligation to resolve the epistemic dispute any easier Chad.

I do not believe this general claim is particularly extraordinary based on the examples I provided.

Depends how you define "God" doesn't it. If you claim you believe your personal god is nothing more than a mental construct, an imaginary friend, I'll take that at face-value and not demand extraordinary evidence. Claims about a creator, judge and punisher of the entire cosmos however are going to require a lot more than just convincing arguments on your part. A lot more.

I maintain that a general epistemology includes other ways of knowing besides scientific inquiry.

Such as?

Mathematics, as I have stated above, is a form of human knowledge independent of empirical verification.

That's generalising it quite a bit, the basis of maths is to count physical objects, seek out real-world patterns and organise them into sets. Mathematics can potentially be and feature elements and calculations independent of direct observation thanks to abstraction, but maths not completely removed from empiricism otherwise it would have no beneficial use in our day-to-day lives.

There must be ultimate and inviolate rules from which proximate rules derive. Presumably, mathematics was true before matter and energy came into being.

No. Mathematics is one of the languages or tools we developed to help us describe and understand reality a little better. It is not a physical entity, its simply a mental construct. It did not exist independent of minds before the big bang, that's absurd. While indispensable its not perfect because it breaks down and ceases to have any meaning when you try to investigate singularities. No, I do not accept "Mathematical realism" as you call it.

The possibility of deity can only be considered if we believe in some kind of underlying order that informs the particular order in which we find ourselves.

We perceive an "order" by looking for patterns, that's how we comprehend and understand things, by arranging the nonsense into something more 'intelligible' and meaningful to us. There is no inherent order in this chaotic ever-changing non-stationary reality.

Subjective experience provides at least one example of something most people consider real that cannot be objectively observed or subject to empirical falsification.

Personal experiences? I don't doubt the person experienced something. They may think its real, except no one else has any good reason to believe or think any of their "anecdotal evidence" is real. They have to appreciate that its simply unrealistic to want their audience to listen to their claims when they put forward such baseless personal testimonies that can't be verified or investigated by anyone else to see if they've misinterpreted or deliberately left out certain details in their stories.