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:
- Only if I had thought that he was sick would I have called him.
- *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.