In order to get both computers working independently of the other, you would need to invest in a device such as a router. This is because in the setup you currently have, software on the first computer bridges the internet so that you can access it on the other computer. When the computer is off the software cannot run, hence the internet won't work on the second computer.
With regards to the email problem, the email software(such as outlook) downloads the email from an internet server. Either the second PC is not set up to download emails from your mail account, or the first PC is downloading them, and at the same time removing them from your server. Either way, you need to make sure your second PC has the email account settings correct in the email client, and that the first PC leaves a copy on the server. This can generally be done fromt the options or accounts menu option in your email client.