1) you can share purchases between kindles as long as their on the same account, but they're not all synced in that way. To send the book to the old kindle you'll have to go to the Amazon website then to Your Account and Manage Your Kindle. Pick the book you want to send and then click the button on the right side and choose which kindle to send it to.
2) yes. I'm not sure it charges as fast, but it works.
3) many people seem to have this problem. I think it's a software bug rather than a problem with the actual screen
4) don't know
5) you can change the carousel by getting a new launcher. I havent don't it but
this thread may help