"Thou shalt not muzzle the ox that treadeth out the corn.", is a profession related storylet that becomes available as part of you receiving An Earnest of Payment.
Every week, you will receive An Earnest of Payment, which you can then turn in for the reward specific to your profession. You also receive Professional Perks, which give generic rewards available for any profession.
You should probably use your Earnest of Payment as soon as you get it, as its rewards can really help push you forward in your Qualities. And while I haven't checked myself, I think that you could safely let your Earnests of Payment stack up, It seems like you can only have one Earnest of Payment at a time and they can't stack, though there wouldn't be a point anyway, as An Earnest of Payment has no use outside of the "Thou shalt not muzzle the ox that treadeth out the corn." Storylet.
Source: http://echobazaar.wikidot.com/profession
As you've gathered, dreams are chiefly long, atmospheric storylines that provide a small amount of orthogonal content. While you don't need to complete dreams to complete the story, the dreams do give you insight into some of the themes and main storylines of Fallen London.
However:
Completing What The Thunder Said gives you the Stormy-Eyed quality, which does turn up outside dreams.
You can trade progress in Dream stories to Dr Schlomo and, in certain circumstances, the Manager of the Royal Bethlehem.
There are a handful of options in the Cave of the Nadir, that place of dreams and forgetting, that are unlocked by dreams qualities.
Dream progress can be hazardous at high Nightmares, as additional Nightmare-inducing cards will appear in your deck.
Death by Water at a specific level also used to be important for beginning the Seeking Mr Eaten's Name storyline, but no longer. Unfortunately, it is now much easier to set yourself on that path, in the Forgotten Quarter.
Best Answer
I just died and recovered; my dream progress is intact.