No urgency needs to be involved, but lopen te … refers to an action that is happening now, and happening for some time. It would be likely that some kind of motion of the seller is involved, e.g. selling things on a fair, or going from door to door, but this is not really required. E.g. I would translate ze loopt te zeuren to she keeps on nagging, there is no link to any motion at all.
Hij is aan het verkopen is happening right now, but this doesn't have the 'this action has been happening for some time' meaning and also not the motion meaning. So is aan het… can have the same meaning, but if you either want to emphasise the 'has been happening for some time' and/or the 'motion', then it's probably better to use the loopt te… construction.
From what I've gathered from this and previous sentences, with verbs such as lopen, zitten and liggen + te, what matters for the meaning of the sentence the most is the following verb, in this case verkopen. My question is, does it matter much whether I use lopen or staan or another one of them? That is, do these verbs still retain their original meaning even in the continuous tense? Does the sentence "De vrouw loopt te verkopen." mean the same as "De vrouw staat te verkopen."? Or do they differ, for example the latter sentence wouldn't carry the same connotation of the action happening for some time as you described?
Using the continuous zit te x and staat te x are clearly linked to the position of the subject, where loopt te x can either link to the walking action or can be more abstract (as explained above).
So if someone is standing and selling something, it can be described as either De vrouw loopt te verkopen or De vrouw staat te verkopen. The former having more emphasis on the duration of the selling action, the latter on the position of the woman.
But if some motion is involved then only loopt te can be used.
It is not connected to the red light district. It is more said about somebody hanging around in a mall, walking around, speaking to people and trying to sell something, like subscriptions to a magazine. Or a Jehoa believe. Walking and selling. Actually the walking part is missed in the official translation.
This is the literal translation, not wrong, but the sentence (zij loopt te verkopen) is used in another context. Usually when people speak to each other they exchange small talk, but sometimes people are lobbying for some idea, plan, concept or political party. The sentence "Zij loopt te verkopen" describes the lobbiers and explains that it is no small talk. It is a little bit of a warning. You can understand what she is saying as you know that she is trying to sell you some idea. The meaning (advice) is: "don't buy that" .