The American way would probably use "would" instead of "should".
It's surprising that he would say that to you.
EDIT: I wouldn't know where to begin to find such a source, but as a native American English speaker I can tell you with confidence that in order to express surprise over the fact that someone did something, the structure is "I can't believe [x] would [y]", "can't believe" being an idiomatic hyperbole for "it's really surprising".
"Should" is pretty much exclusively used to describe something that a person "ought to do" ("You should do your laundry today") or an expected result "this should be accurate".
The subjunctive "Should you need me, I will be here" is considered archaic in American English.
I'm sorry you should forget my birthday.
sounds entirely foreign to me. If they forgot my birthday, I would say "I'm sorry you forgot my birthday". If they haven't yet forgotten my birthday, but I want to indicate that I would be displeased if they did, I would say "I would be disappointed if you forgot my birthday."
Generally speaking if it's regarding an event that did in fact happen, we wouldn't use the subjunctive, preferring a past tense. After all, the subjunctive is for conditional structures. If it actually happened it's not conditional. The only exception I can think of to that is the case I gave above "I can't believe [x] would [y] (implied: but obviously [x] did)".
I'm not entirely certain that I'm reading your first example correctly though:
It's surprising that he should say that to you.
The more accurate analog of this sentence might be:
It would be surprising if he said that to you.