EDIT: I should have probably lead with this, but a quick note that I think relates to a lot of the questions being asked here: we have a single literary reference to the wall, Hadrian's biographer saying that he "was the first to build a wall 80 miles long to separate the Romans from the barbarians". That is the entirety of the direct evidence we have for the function of the wall. Beyond that, all arguments are based on inferences from the structure itself. My personal belief is that, contrary to many modern historian's claims, the structure is perfectly consistent with a primarily defensive function.

While I'm not going to say that it is the most impenetrable barrier, it also certainly wasn't nothing. It is a couple meters wide and a couple meters tall throughout (it varied), which might not stop a truly determined foe but would certainly cause delay. And it would also be effective against raiding--again, a truly determined party could scale it, but they couldn't just traipse across as they please. And of course the wall was manned: as I'm sure you noticed the milecastles may not have housed an army each but they were hardly nothing. I'm certainly sensitive to the more symbolic interpretations but I think that in our haste to ascribe meaning onto the landscape archaeologists often forget that these things can serve practical purposes too. It may not keep out Genghis Khan but that bar may be a tad too high. Another interesting note is that there are ditches on both sides of the wall, so any raiders would have difficulty getting back over the wall.

It is often said in counter to this view that the wall would have been to expensive to justify the cost. When these people produce figures for the cost of the wall versus the cost of scattered forts, the potential economic loss difference between them regarding their differing abilities to protect, and a detailed breakdown of the labor regime employed and how it affected the ordinary schedule of the soldiers in the region, I will take these seriously.

That being said, the more interesting question I think is your last one, what was its position in the sort of social-political ecology of the region? As you know, it was not a boundary between two sovereign states and was thus not treated as such. There are plenty of Roman forts north of the wall, implying patrols and the like. It certainly had symbolic power, the Romans were quite sensitive to that sort of thing, but it isn't as if the commanders on the ground were particularly bound by that (unfortunately I am not aware of any Vindolanda document specifically describing patrols, although I personally believe they are implied in the famous Britunculi letter). This sort of ambiguity is even in what we might call the "official imperial ideology" of the metropole--this is the imperium sine fine (Imperialists Without Borders) that had a god called Terminus.

I'm somewhat out of step with how the wall is usually described--I think in part because "sometimes a wall is just a wall" is sort of a boring conclusion, but there is a somewhat stupefying number of works at least tangentially dealing with it. Edward Luttwack's Grand Strategy of the Roman Empire, an influential and somewhat unfairly maligned book, argued that fortifications served as part of a strategy of "active defense". This idea has had legions of critics, to pick on, CR Whittaker believes that there was no active defense policy, and sees the frontiers in ideological terms. There is also plenty on the wall itself, one I have glanced at being PR Hill's Construction of Hadrian's Wall which makes a pretty solid argument that the wall was build hastily and somewhat sloppily, which I thin supports my argument that it was mainly practical. I believe a pretty standard work is still Dobson and Breeze's Hadrian's Wall, which takes a pretty pluralistic approach to interpretation.

I was pointed here by one of the mods about a similar question I had.

Are there any records of any raiding groups from north of the wall trying to attempting to cross to the south? If so, were they successful or was the wall effective at stopping them?

I would note that walls tend to not be very useful without a garrison, and that the wall's actual use should be compared with the garrison present at various times in the Roman era.