WOW! Lots of questions here, Vuligora! Just FYI, most of these are answered other places (you can use the search function for specific terms.) But I'll indulge you (since I'm stuck on a chapter ending!

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Let's break them down into managable chunks. This will be a REALLY long post, since you asked a lot of specific questions, but hey -- like I said, I'm stuck!
Okay, so you submit to an agent. They like it they take the book on.
Let's break this part out first, even though you didn't ask the question. The very first thing that will happen when an agent takes on the book is that they'll have you sign an "Exclusive Agency Agreement". This gives them the right to negotiate on your behalf with publishers and, in exchange, receive a portion (10-20%, but usually 15%) of the gross proceeds from the publisher for the advance and any future royalties. Now, the first thing you want to do is make sure that the AGENCY agreement is for a specific time. Many of the larger agencies use one year or eighteen months for the term, which is then subject to extensions (either by future written agreement TO extend, or automatic, so that you have to agree to STOP.) It really doesn't matter which form of extension they use, so long as you KNOW what it is and the method of extension is clearly defined. You want to have representation of any future books be OPTIONAL, so that if it's a good relationship, cool. If it didn't work so well, then you're only stuck with them handling the one -- because they get paid for as long as the book stays in print with that publisher. Make sure that if they have an assignment clause (like if the agency gets bought out) you are entitled to notice of the buy-out or assignment BEFORE it happens.
They try to get a publisher interested. So what happens if the publishers say yes. I realize an agreement is made one who gets what percentage of royalties and the rights of the book usually stay with the author. But, how long does the publisher have the right to publish your book? Is it like, we can publish it for three years and no one else can or is it a forever kind of thing.
Okay, before we answer that, let's answer the question that you HAVEN'T asked. What happens if you get a publisher who signs the contract but never GETS AROUND to publishing it? This is the first thing you need to watch for in a contract, because it really does happen. This right is called "Mandated Publication," and it
doesn’t even EXIST in some contracts, so the author is stuck with a contract (maybe with or without an advance) but no book in hand or in the stores. Maybe the publisher is shady, or perhaps the line/imprint it was bought for died before it made it through editing. Regardless, you want to hold the publisher to either getting the book out or returning it. So, an author should consider adding a Mandated Publication caluse to contracts without. "Mandated publication" means that the publisher must publish the book within a reasonable time (often 18 or 24 months from contract signing) or return the rights to the author free of charge. Frequently, any advances paid to the author to that point would be retained by the author as damages for failing to publish. In a multi-book contract, it usually only applies to the specific book not published, and the remainder of the contract remains in force. Red Flag: Watch out for mandated publication clauses that require the advance to be returned if the book doesn’t publish, or other money to be paid for the "privilege" of returning the rights.
Now, how long can the publisher publish it? Again, this depends on the contract, but the clause you're looking for in the contract is the "Grant of Rights." You, the author, are granting the publisher the right to publish the book under the United States or another country’s copyright laws. This right will probably be "exclusive," meaning you can’t sell that same right to more than one entity at a time in the same format, and it will be for a specific time -- either a period of years, or for so long as it's in print. Red Flags: You should watch for language that grants the rights for the "term of the copyright" which is your entire life, PLUS 70 more years after you're DEAD! Also watch for clauses that include the words "sell the work and any subsidiary rights worldwide." A good contract should be specific as to what rights the publisher is entitled to. Whether it is a particular language (English) or specific subsidiary rights (audio, foreign translations, movie rights, etc.), you should know what you are giving away.
Could someone have a publishing company publish their books and then start their own publishing company later and publish and sell the same books published by other companies previously?
This goes back to that whole "assignment" thing. You want to limit this ability as much as possible, but understand that mergers happen. The publisher has to retain the right to consider your book an ASSET of the business. But you, as the author, should force them to keep you in the loop. Make certain that in the assignment clause, there is language that requires the EXISTING publisher to notify you of a merger, bankruptcy, buy-out, etc. What I've seen in a lot of cases is that smaller publishers insert language requiring the BUYING publisher to notify you. But what happens if they DON'T? You don't even know who it is, and the old publisher is kaput -- no address, no phone, etc. You're stuck in a contract without a publisher and have no idea whether it's being sold or if you're entitled to royalties. If you can, try to have your agent insert language that gives you the option to take back the book (and all of the rights) in the event of a merger, bankruptcy or buy-out. For this, they might want a buy-out payment, because they haven't done anything wrong. You're just nervous about the future. You get to be, but they might want a payment, so weigh whether it's worth it to you.
Any other comments by published authors on the subject of submitting to agencies, publishers, and rights, ect. would be appreciate.
I wrote a post awhile back about a whole slew of contract rights and things for authors to watch for. Let me see if I can find the link to it. It'll answer a lot of questions you haven't even thought of yet!
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Okay, here's the link to one about subsidiary rights and what they mean to the author. I haven't found the contract one yet, but I'll keep looking
http://www.absolutewrite.com/forums/showthread.php?mode=hybrid&t=15068