Please read the following points -
- You can eliminate type 3 that 'should' would ever be used there.
- Type 2 is for 'unreal' conditions - so 'should' can not be used in unreal or imaginary conditions
In type '0' we speak of the general truths and the events which happen regularly or frequently. So 'should' can not be used in 'zero conditonal'.
Let us understand this with the following example
For General truths like
- If you heat ice, it melts.
- If you eat too much sweet food, you put on weight.
- If you press this button, the door opens.
Here you can't use 'should' because unlike the examples of type 1 (see type 1 examples below) where 'should' implies probability that something 'could happen', here there is no question of probability in 'general truths'. Because they always give you the same result 'if' you apply them.
If I ever go out, I always let my parents know it first.
In this situation you can't use should because you are not talking about a one-off situation but it is a personal statement i.e. everytime when I go out, I let my parents know it first. So 'should' can't be used in such sentences.
So the only remaining type is type '1'
- If you need any money, let me know.
- Should you need any money, let me know.
- If you have any queries, contact us on the above address.
- Should you have any queries, contact us on the above address.
These are like imperative sentences but only with a condition. Here 'if' can be dropped.
However 'should' can hardly be used in negative imperative sentences.
- If you don't go to the party, come over to my place.
Here 'if' can't be dropped to replace 'should'
I googled to see if I could find any examples of 'should' with a negative condition. But I hardly found anything. So I guess it is safe to assume that 'should' should not be used in negative conditions.
Let's see your examples (the ones from the link)
- Should it rain, there will be no picnic today. (= If it should rain, there will be no picnic today.)
- Should she fail the test, she will lose her job. (= If she should fail the test, she will lose her job.)
In these examples the conditions are not negative, but the results are. If the conditions in the sentences were to be negative, then 'if' would have had to be used.
- There will be picnic today, if it doesn't rain.
- If she doesn't pass the test, she will lose her job.
There could be few exceptions when 'should' is used in a negative condition
- Should it not rain, we will go and water them.”
Please note that here the subject is 'it'. I tried to see if I could make any sentences with any other subject than 'it', but I guess it is not possible.
Even in type '1' there are many conditional sentences in which 'should' can not be used. I guess we will have to develope flair for language to understand it completely.