eCryptFS is not designed for cloud storage. It assumes it is the only application accessing your ciphertexts and runs into undefined behavior when some other application (say your Dropbox client) modifies them.
EncFS also has its problems as explained here.
You might want to take a look at CryFS https://www.cryfs.org
Short answer: Yes and no.
Can root see my encrypted /home folder?
Yes. As long as you are logged in, root as well as any sudo user can see your decrypted files. Also, when you wake up from sleep, your /home will still be decrypted.
Also there is a bug in ecryptfs that prevents unmounting the decrypted /home folder at logout. You should instead shutdown or restart the machine or manually unmount the folder from another sudo/root user. See this question for more information.
Can another user with root privilege change my password, then login my account using the new password see my encrypted /home?
No. Your /home folder is not encrypted with your password, but with a passphrase which is encrypted with your password. Another user changing your password will not affect the passphrase.
At the first login after an administrative password change, you have to mount your encrypted home manually and rewrap the passphrase. For these tasks you need your old and the new password
ecryptfs-mount-private
ecryptfs-rewrap-passphrase ~/.ecryptfs/wrapped-passphrase
When you change your password, the home directory passphrase is re-encrypted with your new password, so you should have continued access to your files with the new password. This is handled via PAM (Pluggable Authentication Modules) (via).
See this related question.
Best Answer
From the terminal type:
If the file exists and contains your home directory path, your home directory is encrypted.
Another way to check is looking at your home dir mount point:
From the terminal type:
An encrypted home will be mounted at the ".Private" filesystem, otherwise it will be mounted at some regular partition device (/dev/*) .
Regular device mount (no encryption):