based more on hunches than on anything actually empirical

That's what abductive logic is.

Clearly he's not being arbitrary. While it isn't purely deduction where the conclusion has to be true if the premises are true, in abductive reasoning it's only "likely" they're true. The better your premises, the better the likelyhood.

And how good are Sherlock and House portrayed as, in this way?

Very.

That's why he's allowed to do almost anything, since he usually ends up finding the right solution despite a little trial and error.

If he constantly turned out to be wrong, he wouldn't have an entire department and there'd be very little point in the whole story

Abduction is basically deduction when you account for reality.