The use of "were- should- had” at the beginning of sentences instead of “if”

The phenomenon mentioned here is often called Conditional Inversion in the linguistic literature. Here's an interesting paper about it.

One difference between Conditional Inversion and if is that inversion is really only possible with those three verbs, as you note, and hence is usually only found with counterfactuals. Another difference is that inversion doesn't work well with the focus adverb only:

1. Only if I had thought that he was sick would I have called him.
2. *Only had I thought that he was sick would I have called him.

Those are my judgements and the judgements of the authors of the paper I linked to, and my guess is that they are in line with modern usage generally, though it would be good to check. The Iatridou & Embick paper gives some more potential contrasts. Their conclusion is that verb-initial conditional clauses can't be focused, and that "The use of inversion is meant to indicate the fact that the truth of the proposition in the antecedent is old [information]".

A side point is that in the recent history of English this type of inversion was possible with a lot more verbs, including could, would, might and did (Denison 1998: 298-300). This might account for the overall rather formal flavour of these examples, as Greg Lee mentioned in his comment.

Still, in a large number of situations the two constructions are completely interchangeable (for me at least).

Ref: Denison, David. 1998. Syntax. In Suzanne Romaine (ed), The Cambridge Grammar of the English Language, vol. 4: 1776-1997, 292-329. Cambridge: Cambridge University Press.

As George said earlier, these inversions are interchangeable and my opinion is that it puts more emphasis on the condition than the if-clauses. I'd say one normally uses it to sound more 'academic', to gain points in school essays and to look good and sound smart. The only thing I would add is that you can use 'should' to show a future possibility as well, not just in the example you gave. For instance, you could say:

Should I win the competition, I would donate the money to charity.

If you want to use an if-clause, you don't necessary need to include the 'should'.

If I won the competition, I would donate the money to charity.

To me, they sounds like an aspiration, a dream, or a possibility (which would be stronger in the first case than in the if clause). I would personally use 'Should I get this...' when I'm thinking there's a good chance that might happen. But it's a personal preference, otherwise I'd say they are completely interchangeable (with the exception of only if, but that's a different type of condition).

Hope this helped :)