I would switch between scenarios. (every significant milestone, or around three or four sessions, but mainly whenever the current "dungeon", "book", or "adventure" is done.) I'd suggest taking a look at the books in the series- there are one or two long term issues throughout the books, but by and large each one is a mystery within the book.
What you need to make sure of is that you finish each adventure before handing things over. The villain might not be dead, but he's beaten for now. The city might not be safe, but this particular threat won't be what kills it. (Note: If the bad guys won, which they do sometimes, then that's an ending too.) In the section on cities, it points out that city aspects are expected to change as the game progresses. Basically, whenever one of those aspects changes, and everyone involved in the hubbub has conceded that there's not much more they want to try, then you can safely change. There are a few possibly hiccups.
First, what was the odd character out doing? In Dresden, you all didn't just meet at a bar, so when bad stuff went down and you all pitched together, why did one of you not show up? (That would be the DMs character.) This isn't a hard problem, but you do need to think about it. Did the bad guys start out by neutralizing them? Are they busy dealing with another part of the problem? Are they recovering from a consequence from last scenario? Or, are they right there besides everyone else? (Some people can play a character and DM the game at the same time. It doesn't work very well for me, but if you can pull it off go for it.)
Second, are there any secrets? Is one the the players related to one of the bad guys? Is the big bad working to find the three pieces of the triforce, one each scenario? Are the bad guys secretly breaking the seventh law? Is one of the players secretly a red court infected? (That one actually came up- A player got infected during a session, and hid it effectively for two scenarios. Then we switched DMs when the DM went for a semester abroad, and I had to deal with this sudden new plot thread, and pretending I didn't know when the DM got back. If you're rotating DMs, everyone just has to be okay with the fact that inside of three or four rotations, everyone is going to know whatever deep dark secret they were hiding.)
The best way to fix this is to just not have secrets, but that takes some of the fun out of it. Depending on the players, they could roleplay not knowing. (I love doing that sometimes- the example in the book goes something like "A ghoul snuck in the back window. You failed the alertness roll, so why didn't you notice?") Thing is, just because you're not letting it get transferred to your character, doesn't mean that it's as fun. I love surprises and plot twists, and that is something that gets harder to do as more people share the DM seat. One way to still have this is to have a bunch of bad guys, each with their own separate plots, each run by a different DM. Then again, maybe the big reveal isn't as important.
Another way, if you're all clever at this sort of thing and everyone is cool with it, is to retcon a little when you need. Anything that happened, happened, but you might try something like "Okay, so the first time it turned out Jim Blackhat was a sorcerer. Then next time in turned out he was also a redcourt, but he had been trying to keep it a secret! But now, in his secret lair, you find the true source of his power- He's been breaking the seventh law!" Basically, add your own twist every time the DM seat changes hands. Just be careful this doesn't get too tangled and confusing, and remember not to change anything that actually verifiable happened. Blackhat fell off a cliff, but survived and is back for revenge is probably fine, we shot Blackhat in the chest at point blank but he was a vampire and regenerated might be, but around the time the players burn the body and douse the unmarked body in holy water, he does not come back. That's why evil villains have apprentices anyway!
Third, all characters get access to all milestones, even if they slept through the whole thing. That sounds a little obvious to some, and wrong and unfair to others, but especially if there are a couple of significant milestones or even one major milestone, you need to give it to all characters. Playing more than a few skillpoints behind, or even one refresh behind, is an irritating shift in power and agency. Your mileage may vary on that, but it's not hard to come up with a story to justify it if you need.
In the end, it works out pretty well. Just realize that you are giving up a certain level of genuine surprise and twist to the story, and the rotating cast will feel a little different than the same tight nit team every time.
Aspects determine what is important in the scene. Those scene aspects defined when the scene is framed are the ones that the GM deemed to be important in the scene; the ones that add flavor to the narrative.
However, narrative games like fate share the narrative direction with the players. In this particular case, though the GM didn't deem the fact that the marketplace was crowded one of the things that would add flavor to the characters' interaction with the scene, one of the players did. And because of that, and because he felt strongly enough about it, he took an action or spent part of his narrative capital (Fate points) to bring it into play.
As a GM, never feel that you have to define everything- only those things that are important should be defined before hand. And as a player, never think that you are confined by those things the GM has declared- Fate points and your skills give you the tools that you need to help to expand and drive the narrative.
To directly answer your final question, i.e. "I understand that Create an Advantage is about not only creating aspects, but also getting a free use. Does one have to Create an Advantage to take advantage of aspects that seem implicit, but were not yet declared?"
Creating an aspect is one way of getting the use of an implicit aspect. In this manner, you get a free invocation. However, if you're not looking for a free invocation, i.e. you just think that it's important to note that the marketplace is crowded, you can suggest it freely, and if everyone agrees, the Crowded aspect is added. This is stated in the Fate Core Rulebook on p. 78. It can also be found in the Fate Core SRD- Creating and Discovering New Aspects in Play.
Then later, if you, or someone else, wants to use it, a Fate point can be spent as normal to invoke it.
Best Answer
There's a distinct demarcation in games between the Player and the Character. And in most games when such things come up, it's relegating the player to the same position as the character- and trying to force the player to solve problems is if he is the character.
There is nothing wrong with that approach, in any game. And there's also nothing wrong with the other approach, either.
One thing that I take into consideration when playing Fate, however, that makes me look at things in a different light is that even less than other games, you're not playing in order to win or even to solve the problem. You're playing to create an interesting story.
It's for that reason that anyone can suggest a compel. And the verbiage on that point is very specifically stated on FC71, i.e. "if a player wants to compel another character, it costs a fate point to propose the complication." As the player sees something interesting that is instrumental to what they want to play, the player can compel someone else's character. That, I think, is an important distinction, and shows what the designer's intent was towards the interaction of players and characters.
So, taking that further towards your own situation, that demarcation between player knowledge and character knowledge only exists in terms of the narrative itself, i.e. if the player suggests something that is dependent on knowledge that the player has- and not the character he is suggesting to has, then there's a distinction that needs to be made for the narrative's sake. But other than that, anything's game, and not only can they suggest, but they can actually affect the other character if it's important enough.