Show Inbox when starting Thunderbird
From MozillaZine Knowledge Base
Usually if you go to "Tools -> Account Settings -> Server Settings" and check the box that says "Check for new messages at startup" Thunderbird will start in the Inbox folder. If you have more than one account, Thunderbird will start in the Inbox of your default account, which is the account displayed at the top of the folders pane. To make another account into your default account, go to "Tools -> Account Settings", select the account you want to make the default, and click the "Set as Default" button.
However, that doesn't always work in recent versions. There are a couple of add-ons you can use as workarounds.
If neither add-on works around the problem it might be because you changed mail.server.login.at.startup , which defaults to true. 
Opening some other folder
You can also specify any folder as the startup folder (doesn't have to be an inbox) by changing the command used to start Thunderbird. For example, the command might be in the properties of an icon on your desktop. At the end of the command, add
For example, here is a command that starts Thunderbird on Windows, opening a Templates folder:
"C:\Program Files\Mozilla Thunderbird\thunderbird.exe" -mail mailbox://email@example.com/Templates
You can use this method to make a collection of icons that directly open various folders.