I want to create a public (password given out to everyone) gmail account that anyone can use (for subscriptions, using useful websites that require your real email but spam you, etc.) and that's the easy part (I think anarchy might erupt in the account at some point). The hard part is implementing a secret password only I know that is used for changing the account's password. This is to prevent people from abusing the system by changing the password so that nobody else can use it. How can I implement this in gmail? If it's not possible in gmail, how about elsewhere?
You can do this in gmail by using application specific password. Drawback is that the users cannot use webmail but instead must use desktop mail application. However, the problem with having a single, shared junk mail address is that everyone will be able to see everyone else's emails. This is usually unwanted, due to privacy reasons (I don't want other users of the email to know what sites I signed up for) and volume reason (I want to see my own registrations, instead of having to trawl through of hundreds of other people's junks).
An actual throwaway/disposable email provider like mailinator.com would be more suitable for this purpose than what you're proposing.
What you want to do makes no sense as far as I can see.
It's also not possible with gmail.
How about using a service designed for exactly the use cases you mention, such as https://www.mailinator.com/ ?