In order to have access to the Trezor seed, I could store it, using Trezor Password Manager, in an encrypted file in DropBox.
I would never suggest doing this.
You can't be 100% sure whether:
1) the implementation of the encryption is flawless
2) their server don't get compromised
What you are doing when backing up via online services is: You trust a 3rd party.
Cryptos are made to be functional in a trustless system. Don't rely on someone to keep your backup.
If Trezor is as safe as reported, then it should be impossible to get the seed unless one already has access to the physical Trezor device and the pin number.
There already has been a vulnerability which allowed people with physical access to the trezor to get the seed out of the device (without pin).
Hardware wallets are safe, yes. But not 100% secured. Nothing is 100% secured.
That may therefore sound as if it has no adantages, but I can think of several scenarios when it would come in handy:
1) Your paper seed backup has been destroyed.
2) The paper seed is unaccessible due to whatever.
3) Or worse, the paper seed has fallen into the wrong hands and 'they' will sooner or later figure out what it is.
In all of these scenarios a second (paper) backup in a different place would be the solution.
No need to upload your encrypted private key into a cloud.
With physical access (and known pin) to your own trezor you could simply just create a new seed and send all of your coins over to your new (non-compromised) trezor seed.
If a (paper) seed backup gets stolen, simply create a new one and send your coins over (since you have physical access). No need to backup your seed in your trezor itself.
If you can access your trezor, you can access your coins. So whats the explicit reason to store the seed inside trezors password manager?
And can you see any problems using the Trezor Password Manager to store passwords / addresses or seeds for other crypto currency wallets (Those not supported by Trezor)?
This actually (storing the seed of cryptos not supported by trezor
inside trezors password manager) does make sense.
I would never suggest to store private keys (or any other sensitive information) online.
Every encryption can theoretically be attacked. Quite a lot keys already got compromised because of bad implementations (e.g. bad pseudo random number generator).
You can never be 100% sure this won't happen with the encryption algo you are going to use.
My advice would be to keep all of your private keys
offline.