"Should" is a modal verb which, among other things, indicates the conditional mood in formal English. I don't believe it implies a stronger urge or force. There is no real difference between "If you should have to kill him", "If you have to kill him" and "If you need to kill him" except the first one automatically sounds like it's being said in Received Pronunciation accent (because it sounds formal).
"Should" lets you do something else though, it lets you drop the "if" but keep the clause conditional. So you could write "Should you have to kill him" and it means the same as the other options.
One thing I want to point out. "Should" isn't replacing "need" in those sentences, "have" is replacing "need". I think "Should you need to kill him" is grammatically fine, but doesn't sound quite right to my ear.
I think your second example sentence is doing something different. Here "should" isn't indicating a conditional, it is indicating an obligation or advice. It's equivalent to "I strongly recommend you talk to the teacher". This is not a formal usage, so I think it would sound fine in informal English. "You need to talk to the teacher" might be a more common way to say it though.