I notice that even relatively new LPS lamps have huge black socks around their electrodes. Is this something particular to LPS? Or does it happen equally with fluorescent and we just can't see it behind the phosphor?

Here's what I have heard. In the factories for SOX lamps, they used a sensor of sorts in order to ensure that the getter applied inside the arc tube was even on each side of arc-tube. In later years, this sensor ended up breaking, the result was uneven coating of the getter inside the arc tube, often with one electrode getting an absolute sock while the other only got a small amount.

There is a getter in the arc tube?!?! This is news to me! What is it made of?

I did a bit of research, and it turns out it is not a getter; it is excess Barium that evaporates at some stage in the arc tube filling process. Everything else I said is correct though.