Is it possible to only remember the master password to the password manager and still be safe?
Depends on circumstances. If you assume that nobody will ever guess, crack or otherwise get your master password, then sure, you are as safe as can be.
The problem arises if somebody does get your master password - then it's a much larger damage than if you were remembering different passwords.
It's simply "keeping all eggs in one basket" - it might just be too much for your level of paranoia.
For example you must also know the password to log in to the computer (and possibly the password for full disk encryption) before you can access the password manager. Multiply this for each device.
Not an issue. The safe itself needs to be encrypted, then you can and should replicate your (encrypted) safe wherever you need, possibly even in public spaces (of course only if you believe that it is resistant to brute force attacks and free of bugs and backdoors).
For example, I have my KeePass safe on my home PC as well as on my smartphone; especially the version on the smartphone can easily be accessed by whoever gets his hands on it. I believe the password and the program to be "safe". I would never put it in a public space (e.g., on a github repository) because while I believe the software to be clean now, it would not be the first time that software turned out to be bad after the fact.
This is a bit of a different issue though, "do you trust your password safe". I assume this is given, or your current question would be moot.
If you backup your password database then you probably need a password for it too (especially if you’re using a cloud based service, such as Google Drive or Dropbox, for the back up).
You would never have a cleartext password database in the first place, so this is not an issue. See above, it's the same thing.
Also if you have multifactor authentication enabled, you probably need to remember the password for your phone too (if using app such as Google Authenticator).
Sure, you need to remember passwords for the devices you actually use to access your safe. Nothing will ever change that, unless you use the same password for your safe and your devices. This would, at least at my level of paranoia, be unacceptable though since it would make it more likely for it to be intercepted.
Is there a better way, or am I misunderstanding something? Is it really possible to only need to know the password to the password manager?
In principle, yes, with the exception as above.