Should certain sites specifically encourage askers to tag with location?
YES! We should. But I'm going to answer the rest of this from a (purely) IPS point of view, as that is the other site I moderate.
We've got comments for encouraging location tags, and IPS encourages use of those tags by including a few lines about the importance of cultural background in our 'How to write a good question' FAQ post on meta.
If IPS gets questions that depend too much on a cultural background, for example, questions about etiquette or customs, they are generally commented on and may end up closed for lack of details about cultural background. That's another way of 'encouraging' people to include their cultural background (or location).
Note that I said 'cultural background': IPS does like location tags, but we like cultural background even more. There's so much people that live in The Netherlands but that have a different cultural background than me, that a location tag might not be as helpful, it might give less information about this than a line or two in the question. And that's only The Netherlands, bigger countries might have even bigger differences.
Should we have some small popup or other attention-grabbing device to suggest to question askers on specific sites that they should tag their question with their location?
I don't think it'll help. A pop up is too much, too invasive, it might scare people that value their anonimity away. And it might confuse users that ask questions about stuff that's not location related.
People ask on sites like TWP or IPS with very personal questions, and would often like to maintain some anonimity while doing so. So they sometimes don't want to include their location, out of fear that people might trace their question back to them.
Besides, not every question needs a location tag. We've had questions on Social Penetration Theory and the Interpersonal Theory of Depression for example, that don't need one. Having a standard pop-up/attention-grabber suggesting the use of a location tag where none is needed is only going to be even more confusing.