I don't have all the answers, having played quite a while ago and not having finished all the content.
Ambitions are definitely unique and take many steps to go through. Such steps are sprinkled throughout the game, across the span of skill levels you will acquire.
Basically, your first choice will give you access to a number of Storylets that start this particular storyline with beginner-level tasks. Once you're done with this, the next step will have a task requiring a much higher skill level, so you will have to go through a good deal of the other content before you come back to it. At some point, the "next step" will require access to restricted locations which, again, requires a certain skill level or acquiring unique items.
You can cancel your Ambition by going to the NEX tab and buying the Nethean Tea Leaves. This costs 50 Nex, which means it will quite certainly cost you real-life money. I have no idea if doing this once done with a first Ambition (most likely at the "end" of the game) allows you to start anew with a different one. It's possible, but be aware that your skills would be such that you would most likely breeze through it, which may dampen enjoyment.
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
For each of the Primary Menaces (Nightmares, Scandal, Wounds, and Suspicion) there is a consequence upon reaching level 8 with any of them.
For Suspicion, once you reach Level 8, you are hauled back to New Newgate Prison, where you'll be stuck until you reduce your Suspicion to 0.
Before that reach that point, however, you will notice some other events related to high Suspicion.
For instance, once your Suspicion reaches 4, you'll see the "Find a way to throw the Constables off the scent" Storylet appear at your lodgings, and give you an opportunity to ask your friends for help.