Should I send a letter of recommendation directly to a student?

In general, this is a generous thing to do, if you are comfortable with the student. I would make sure to ask what school/program it is being sent to, and then you can double check (and decline if it's fishy). You can ask this question without acting suspicious, just say you want to tailor the letter for the position.

I have had to include my letters for a few postdoctoral positions I applied for (in Italy and Croatia), and I cannot be more grateful for the fact that I had some senior researchers who were willing to send the letter directly to me so I could include it with my application materials.

I've never heard of it being a requirement, but sometimes it is recommended that you include all application materials as one packet. For instance, once when I was applying for a travel grant, it was highly recommended that all materials be put together so that everything arrived on time. However, they required that the letter was on official letterhead and that it was sealed in an envelope and signed across the the seal by the teacher giving the recommendation. This ensured that it was indeed an original copy and not tampered with.

I'd check the web site of the school in question. In the absence of an answer, write to the registrar. If all else fails, decline, because I believe (without much evidence other than my own feelings) that schools give less weight to recommendations not confidential from the student.