One category of mathematical result that belongs to 1 is statements that you need to know are true and that have complicated proofs. Obviously some such proofs are worth knowing because they will help you find other, similar proofs. But not all of them fall into that category. For example, almost all mathematicians can get by just knowing that it is possible to construct a complete ordered field. And perhaps a more important example: many mathematicians use Lebesgue measure, but all most mathematicians need to know is a few basic facts about it, and not the full details of the construction and proof that it works. Another result I remember my undergraduate lecturer more or less explicitly apologizing for was the simplicial approximation theorem, which I remember disliking intensely.
Why do we teach results like this? One reason is that when we teach we are not just equipping people with the tools they need for research, but also demonstrating that we can build up the edifice of mathematics from just a few basic axioms. One can argue about whether we really do this, but I think we tend to do enough to convince any reasonable person that it can in principle be done. If we were to start leaving lots of gaps (there's this thing called Lebesgue measure ... it has the following properties ... it can be shown that these properties are consistent but the proof is tedious and I'll omit it) then this valuable aspect of a mathematics course would be in danger of being lost.