conditional constructions - Is there a difference in meaning among 'should', 'happens to', or 'should happen to'?

What is the difference in the meaning of following sentences (I'm asking for detailed anlaysis of these sentences)

1. 
   

   If it rains, I will stay at home.

   
   


2. 
   

   If it should rain, I will stay at home.

   
   


3. 
   
   

   If it should happen to rain, I will stay at home

   
   


4. 
   

   If it happens to rain, I will stay at home.

   
   

   Can I say 'there is no difference in the meaning among 2 ,3 and 4 sentences ? There is no difference in the meaning and usage among 'should', 'happen to' or should happen to' together. Do they all mean 'by chance' ?

   
   

Again , i would like to know if the following sentences are grammatically correct or not:

1.If she should happen to read the newspaper, she would see your article.

2.If they should agree to come ,we would be delighted.

Source :perfect English grammar.com in Advanced conditionals Explained (pdf )