George,
I assume that your statement 'that faith may be interfering with physics' alludes to the perceived irreconcilable relationship between faith and reason, by which I am interpreting your interpretation of 'faith' as being religious convictions.
But the term 'faith' has much wider, generally applicable meanings; that of conviction, trust, reliance, assurance, belief, devotion, loyalty, etc. that are often applicable to scientific theories that have not been established and endorsed as 'true'. Indeed what scientific knowledge is unequivocally certain and correct for all time? Your point is well taken when you state 'There are some conundrums in mathematics, however, that will never be solved'; not withstanding our understanding that 'never' is a long, long time. As you have further stated; 'the incompleteness findings apply to every branch of mathematics.' From a practical point of view we need not seek perfection in terms of the absolute truth but rather proceed until we have achieved tentative conclusions that have utility values.
Concerning your question 'How does intelligence emerge from unintelligent components?'; your answer preceded your question; i.e. that 'all interesting structures' ... 'exhibit increases in order, structure and variety quite at odds with the Second Law' 'All' is a dangerous word to use but I get the gist of your point - that evolution compounds complexity.
The notion that '... anomalies are all explainable within the laws of nature' is more succinctly posited if we substitute the word 'principles' for 'laws'. Strict laws are absolute while principles accommodate deviations within limits.
To state that there are 'no' non-physical causes the 'no' cannot be confirmed (much like 'never' and 'all' referred to above). These terms should not be used if we are trying to convey the truth as we know it. If there is a God he (she or it) may well call you to account unless he elects to expose himself physically.
The issue of a single universe vs. the multiverse theory is not going away soon. I think that you are correct in suggesting that 'the choice boils down to an ideological one - what are you comfortable in believing?'
Your conclusion is exquisitely stated.
Thank you.
Gary.