Alright, here goes (I haven't been to bed in some time, so bear with me.)
Bastsan: Rather than the underground, he is a god of commerce, but focuses on the raw aspects of it. Wood and food are abundant and strictly speaking, worthless in a DF world unless tended to by expert chefs and craftsmen. But precious gems have value due to rarity, and minerals such as iron, gold, and silver ore have value in utility. A fortress builds its wealth on the minerals it has access to - a town or fortress with naught but copper and lead is going to have to work much harder than one blessed with
iron and gold to generate the same level of prosperity without resorting to prepared meal barrels, if you'll pardon the meta-ness of this.
Thunder comes with storms, and bad weather makes for poor trade, as harsh winds and rain can make wagons sink or lose traction or batter apart vessels large and small.
Gogol: With suicide comes war (as veterans off themselves at disproportionate rates to the general populace,) death, sacrifice, and chaos. In this circumstance, Gogol can be interpreted as a a god of war, but a defensive one - the most ardent followers of Gogol "killing themselves" so they may better protect their countrymen and carry out the will of the crown by way of military training and indoctrination. As such, Gogol probably also holds dominion over those other aspects, with suicide being given a creative interpretation - and this creative interpretation has lead to his common perception as holding dominion primarily over suicide, despite being the de facto god of so much else.
However, these are all still heavily negative, so it's likely only professional soldiers and their families give him any serious consideration in their spiritual affairs, and some may even hold foreign mercenaries in somewhat low regard, believing they should return to thier own countries, not become embroiled in the affairs of the the pearlites.
Sarrak's take can also mesh with this fairly well.
Edri: Association of labor, crafts, and healing likely stems from the shepherds who had to tend their injured flocks and families themselves, and those flocks providing much needed thread and cloth for mending the wounded. If you didn't have a surgeon to fix the wounded, then the frontier folk would turn to the people who tended the sheep, llamas, and alpacas, or to the weavers and tailors, since they were the next closest jobs associated with the most vital of medical supplies, thread and cloth.
Thus, the labors of healing, weaving, clothesmaking, and wool production become inextricably linked in the minds of the pearlites.
Pasmug: This is another one that requires some creative interpretation, but Pasmug wields jealousy as a force of self-betterment, and by extension the betterment of the nation. If you're jealous of what your neighbor has, you must either work harder, or accept it, and by working harder you not only better yourself, but you better your own home and family as well, as your increased skills in whatever you do, means work is done faster, and goods are produced with greater quality, abundance, or just plain haste, and that's good for everyone. Besides, my observations indicate people tend to value skilled workmen over haulers and rookies.
Envy may be a deadly sin, but it needn't always lead to destructive practices.
Sabu: Justice often has little to do with fighting, but it could have something to do with ensuring things are kept in fair balance - you overfish, you lose a food supply. You don't enforce the law, you lose people to migration or riots. It's an odd combo though, but that fishermen and sailors pay homage to him is probably a good assumption, and may even do so in equal measure to Bastsan, praying to one that the fish are plentiful (and don't try to kill them,) and the other for safety in storms during crossings of the blue.
Ino: I'm unsure what to really do here. Oaths and loyalty don't really seem to jive well with agriculture, where with fishing and justice one can take an angle of maintaining balance.
EDIT: Some of them required that creative interpretation, and I'm fairly sleep deprived, but I hope they don't sound like the ravings of a mad man.