Please see screenshot below. For every product I must check for Main Website even if there's only one website. Is there some way to have it checked by default?
Your screenshot isn't available.
Can you share the same screenshot and another one form System -> Manage Stores ?
If there is only one website and store as shown in the screenshot, then the website tab shouldn't appear on the left unless you have switched multi-store option. Usually whenever there are multiple sites to list products then this screen should be used whereas in your case this should not even been shown.