I'm a regular employee (of the UK presence; it's all EU, so I assume that makes things a bit easier). There's a German tax office that does the wage accounting to make sure that it follows all the rules.

The only difference to a standard German employee is this: While in Germany legally the employee owes the wage taxes, they're usually collected by the employer and passed on to the authorities. But the German tax authorities don't like to do that with foreign companies (basically because they can't hunt after them when they don't pay). So from a legal perspective, I pay the wage taxes myself (but the tax office takes care of that as well).

I never knew that was possible! I'd have imagined it could have been possible as a contractor, but not with this arrangement. Is there any particular benefit to being an employee in this case as opposed to a contractor?

I expect some aspects of EU working law apply, which gives you a lot more protection than a simple contracting relationship.

I may be wrong, but if they have a presence in the UK they most likely do not need a presence in Germany. Through EU agreements, Germans are allowed to work legally in the UK.

They are allowed to live, work, and study in the UK without any significant restriction, and the tax issue would be a non-issue if they were domiciled in the UK. The reason it's interesting and complicated is because they're apparently living and working in Germany as an employee, without FogCreek having a German entity. They wouldn't be subject to British taxes and there's no German business entity to register with the German tax office.

A UK company (or a company of any jurisdiction) can establish a subsidiary in another EU state, in much the same way you could register a NY-incorporated company as a foreign company in CA, but this doesn't appear to be what's happened in this case.