(Correct me if I'm wrong.)
You're wrong. You can enter the UK for up to six months per visit without a visa. In all other EU countries except Ireland, the limit is 90 days in any 180-day period. This is counted separately for each non-Schengen EU country and for the Schengen area.
Ireland is either 90 days or three months; I don't remember which. I think the limit is per visit rather than within a certain period, but I am not certain.
This is presumably allowed. (If not, why not?)
It's certainly not allowed in the UK. The reason is that the law forbids people to work in the UK without a proper visa. The other reason is that legislation and even political consensus haven't caught up with the realities of modern technology.
(Canada, by contrast, explicitly permits this on the argument that such work does not affect the domestic labor market. Whether other EU countries are more like the UK or Canada is not particularly clear.)
Do I need to mention any of this if I somehow get stopped on entrance and asked what the purpose of my visit is? I'd like to know if the one-word answer "tourism" is sufficient in this case.
If you're going to the UK, failing to mention it in an interview would constitute deception, a serious violation that can lead to a 10-year ban, but mentioning it will result in refused entry. A conundrum.
However, US citizens these days are not generally interviewed, thanks to automated passport gates, so you probably won't have to employ deception. Working during your visit will still be a violation, howesver.