Well, let's test this scientific theory of yours by asking for examples... Please, list such "intrusions"..
Well, there's the Bible - the people who originally wrote it existed on the material plane. They also needed faith to believe that the stories (or in several cases voices in their heads) were real. Therefore faith has had to intrude on the material plane, if it didn't we'd have no knowledge of it.
The laws and traditions deal with ethics and philosophy, not scientific discipline(s). I wonder, why you even brought these up.
Probably because in the past (and present), religion has deemed to weigh in on scientific arguments. I hate to bring up the cliche that is Galileo, but as foolish as he was, he (and others) had their scientific writings banned by the Catholic church.
Citations, please [regarding great detail]
Granted, it's not great detail, but they do claim to understand why god did things - generally it's because he's angry with his creation for not behaving the way he knew they wouldn't.
Citations, please. [regarding falsifiability]
Talking serpent, great flood...
Personally, it's not the faith in a supernatural being that I find bewildering, it's the faith that said omniscient being is good and that it didn't create humanity for the sole purpose of trolling them. Read the bible without the view that god is good and you realise it describes us being set up to fail from the get go.