The short answer to this is that the small folk do not have surnames. As George R. R. Martin says himself.
Bastard names are given only to bastards with at least one parent of high birth. So the bastard child of two peasants would have no surname at all.
Thus a bastard name like "Snow" or "Rivers" is simultaneously a stigma and a mark of distinction. The whole thing with bastard names is custom, not law.
So Spake Martin, SF, Targaryens, Valyria, Sansa, Martells and More
However, the whole thing is a bit more relaxed than that, as can be seen above with bastard surnames being custom not law. As such it's likely that "rules" around noble surnames are more of a custom and not law though likely held to a higher degree of scrutiny given the weight they carry.
For what it's worth Janos Slynt was definitely a Slynt before becoming Lord of Harrenhall and founding House Slynt. Though he is already the commander of the City Watch at this point.
Lord Renly Baratheon was less sympathetic. "If you cannot keep the king's peace, Janos, perhaps the City Watch should be commanded by someone who can."
Stout, jowly Janos Slynt puffed himself up like an angry frog, his bald pate reddening. "Aegon the Dragon himself could not keep the peace, Lord Renly. I need more men."
A Game of Thrones, Eddard VI
I haven't been able to tie anything down to why this was the case so it could be any one of the following (speculation though of course):
- High ranking members of the City Watch can use names
- Names are custom not law so as long as no one questions it you're fine
- GRRM goofed
As for Allar Deem the same sort of thing is likely considering he was Slynt's "right arm" and so high up again.
"Truly." Tyrion filled Lord Slynt's cup to the brim. "I have been glancing over the names you put forward to take your place as Commander of the City Watch."
"Good men. Fine men. Any of the six will do, but I'd choose Allar Deem. My right arm. Good good man. Loyal. Pick him and you won't be sorry. If he pleases the king."
A Clash of Kings, Tyrion II
Another option for this appears to be that there is a middle class in Westeros that aren't quite Noble yet but are also not common folk. This would appear to match with Slynt and Deem but there is no concrete case for this. It could also be that they just use the last name as a nickname or that no one questions it of course.
As for Jeyne Poole, well, House Poole are a noble House of the North.
The largest banners were behind the dais, where the direwolf of Winterfell and the flayed man of the Dreadfort hung back of the bride and groom. The sight of the Stark banner hit Theon harder than he had expected. Wrong, it's wrong, as wrong as her eyes. The arms of House Poole were a blue plate on white, framed by a grey tressure. Those were the arms they should have hung.
A Dance with Dragons, The Prince of Winterfell