I have three, 7, 5 and 3. They do fight. They're allowed to fight up to a point, especially in winter when they can't play outside and they get a little riled, but they aren't allowed to bother me about it. When I say up to a point, I mean if screeching and crying commences and I say, "Stop" they know they have to or discipline will ensue. As for bickering over toys and stuff....they're allowed to amongst themselves, BUT if anyone starts to cry or comes to me and says "he took my..she won't give me..it was my turn..." My response is something like, "Do you need me to take it away and would you rather clean your room than play?" so they usually keep their disputes to themselves.
Yes, they have their "own" stuff, and no they don't always have to share..but again, I just keep excessive fights in check by enforcing warnings when they've been warned to pipe down. They're not allowed to act like that at all with other kids, or when I'm exhausted and not in the mood.
Basically you just need consistent discipline in all areas of your household so your kids listen when you give an order. That way you can keep the fighting in check to the level that you can handle at any given time. If I need them to be REALLY well behaved, like we've been out having fun all day and I need to sit and do the bills in peace when I'm exhausted, I'll give a warning like "OK, you need to play nicely, the first person to shriek or fight will get a consequence" and that does the trick.
Don't let them manipulate you guys with the books or things like that...just say, "It's ____'s turn tonight, no fighting." And then enforce with discipline if necessary until they get it that they aren't allowed to fight over books. The firmer and more concise you are with night stuff the better since they're tired and easy to upset. I never witheld books, but I did "warn" (threaten spanks but never had to do it because they knew I meant it) them to sit nicely for books because I knew that books were going to be important at bedtime for years, so they needed to just get over the bedtime book behavior challenge immediately. It's been great because they all love books at bedtime and now my oldest reads to herself, but they were squirrelly before I put my foot down which is normal.
As for TV, they hardly ever get to watch it, so the first time I just cruised by and shut off a long awaited show and sent them to bed when they were fighting over it pretty much handled that. Now I hear them whisper-fighting over shows.