"That there should be" is more of a suggestion. "For there to be" is more of a statement, and sounds more certain. But they are so similar, you can use them interchangeably in most cases, like your examples. In your examples, I'd say "for there to be" is the better choice because of the words "vital" and "important." Here is an example where "That there should be" is best. I think that there should be more support for school teachers. Here is an example where "For there to be" is best: It's important for there to be enough chairs at the party. If you are making a suggestion, and it's a soft suggestion, use "there should be." If it's a firm suggestion or statement of opinion without much room for debate, you say "for there to be." "For there to be" is always started with an adjective, like "It's important.../It's vital..." etc. But "that there should be" can start with "I think..." which lets you know it's a soft suggestion. Hope this helps!

I'm no grammar expert but I think that your question comes down to grammatical mood. The meanings are extremely close and others may have better explanations. Version 1 is present subjunctive and is less definite. To me it feels more formal. It is a softer way of saying it. To me it feels a bit old fashioned but it might be appropriate in formal contexts. The subjunctive is replaceable with the indicative in a lot of informal uses of English. Version 2 is indicative, it is simply stating a fact. This feels more natural and informal to me In this case you are directly saying what you think. https://en.wikipedia.org/wiki/English_subjunctive https://blog.oxforddictionaries.com/2011/02/10/does-english-have-subjunctive/

They are completely interchangeable. Also, you could write “It is vital that there be . . .”, and that would be interchangeable too.

Both are correct, but I prefer (b) in both. The word "should" repeats the meaning of "vital/important" in (a), so it's not great style.