I have to say that I think Debby's advice is INCREDIBLY solid on this front. Novels change SO much in revision that if you write book two before you sell and revise book one, you stand a good chance of (a) not selling book one and then not being able to do anything with book two anyway, or (b) selling book one, getting editorial input, completely changing some of the characters/plot/ending in key ways, and having to throw out what you'd written for book two and start from scratch. I'm only saying this, because I've been there and done that for both (a) and (b)! Before I sold my first book, I wrote a series of four books that ended up being good practice, but not commercially viable. I don't really mind or regret having written them in any way, but when you write a series, it either sells or doesn't as a whole, and in a business where it can take MANY different projects to break in to the industry, sometimes the difference between moving on to a new project versus working on a sequel can be the difference between writing seven books before you sell one (which is what I did) and writing two.
But even if you do sell the first book, it's likely that you'll underestimate the number of things that could change in revision. Of my published books, there are three sets of two (three books and their sequels), and the process between revising book one for my editor and writing book two looked a little something like this:
Golden- I wrote the book AND a sequel before I sold the book. My editor was excited from the get-go about the possibility of doing more books set in the same world, but her revision suggestions really shifted the focus of the first book so that it concentrated a lot more on something that was a much more minor theme in the first draft (cliques), and really upped the importance of certain characters and plot elements in the first book, to the extent that when it came time to writing book two, even though my editor said "It can be about anything you want, so long as you call it Platinum," there was NOTHING in the first sequel I'd written that worked with the revised first book. So I threw it out and wrote a new sequel completely from scratch. In terms of your other questions, for this particular set, the second book picks up 2-3 weeks after the first one leaves off and switches narrators. I'm fairly certain that most readers would have preferred a book that stuck with the original main character and concentrated on the one issue (romantic pairing) left open-ended at the end of the first book, but I felt like it was really important to tell the other side of the story...
Tattoo- For this one, I wrote the first book as a standalone and didn't ever intend for there to be a sequel. The ending of the first book ended up changing entirely in revision, and there's a twist in the last chapter that was actually my editor's brainchild- and THAT twist ended up being the basis for the book's sequel, FATE. So if I'd written the sequel before the first book was ready to go to print, I probably would have had to throw it out, and I worry that I might have been more reluctant to really gut things in revision. When I did finally end up sitting down to write the sequel, I had it in my mind that it would take place maybe six months after the first book, have the same tone, and follow a similar plot arc, just with new baddies and everyday traumas replacing the old ones. Annnndddddd... this did not work. At all. I started the book four or five times, before I finally realized that I was basically trying to write the same book again, and that I needed to give myself license to make the sequel different. The result ended up being that I started book two a full two years after book one ended, and the focus shifted from the theme of the first book (which is very much so "friendship") to focusing more on the main character outside of her friendships.
The Squad: I sold this one in a two-book deal, so I knew up front that there was going to be a sequel. Because of overlapping deadlines (the two books released on the same day), I ended up writing the first draft of the second book after having done only one revision on book one, and (though I know some writers who enjoy doing this), it drove me a tiny bit insane, because right after my editor read book two, she decided that she loved the twist ending and that book one needed a big reveal in the last couple of chapters, too, and that, ideally, it would be a twist that would play an integral part in book two, even though book two had been plotted and written without aforementioned twist in play. Everything worked out, and I'm super happy with how the books came out, but it was definitely a challenge at the time! Oh, and in answer to your other question, book two started three weeks after book one, same characters (and same relative level of focus on each of those characters). These were very episodic books, so they were structured kind of like episodes of a TV show- there was a "case" in each of the books, and that case was closed and all plot lines related to it resolved by the end of the relevant book, but some more general elements (the character's relationship with the main love interest, etc) stretched from book one to book two before they resolved (or came closer to a resolution).
Writing sequels is an amazing amount of fun, but it can be tricky business! I have a good friend who says that as a writer, you never learn how to write books... you just learn how to write the book you're working on at any given moment. I feel like that is especially true for sequels- every time I write a sequel (every single time, I swear), I have to figure out all over again how one goes about doing it- how the second book is similar to the first, how it's different, how it's *okay* for it to be different, how different is *too* different, how similar is *too* similar... and when you're having to balance all of those different things ANYWAY, I feel like it just adds a lot of complication to sit down and write the sequel before you know what the first book is going to end up being if and when it goes to print. Even a change that seems small (say, your editor wants to see more character growth in your main character, or wants you to combine two of your supporting cast members) could completely change the sequel- if your character ends up in a different spot emotionally in the revised book one than the first book one, then their entire arc for book two could be out of character or redundant; if you get rid of a side character in book one and it just so happens that your original book two was largely concentrated ON that character...
There are just a TON of variables, and while there are some advantages to knowing where you want to go in book two, I think that can also handicap you, because if you write book two, it might impede your ability to really, REALLY revise book one in conjunction with an editor to make it the best book it could possibly be.