And the certificate was encrypted by the database master key and not a password?

Yes it was encrypted by the DBMK because it was not encrypted with a password.

If you export the database master key from the original server and restore it on the new, can you open the database master key manually and then everything works?

No, I open the key by using the following command:

OPEN MASTER KEY DECRYPTION BY PASSWORD = 'pwd'

and then I try to open the symmetric key and it gives me the error.