OakDryad, I don't believe there are any servers that cannot be compromised. However, complex passwords and two factor authentication are the best ways to ensure that you keep your data private. Also, if you use a password manager to log you in, you will not activate a key logger. A password manager also takes the worry out of remembering passwords if you choose to log out.
There are several types of key loggers, but some are based on hardware to capture your keystrokes and others are apps that you are tricked into downloading. The normal warnings about being careful when 'clicking' links applies. Having good anti-virus protection is essential.
As to whether or not you should sign out of websites, if you are confident that no-one you don't trust can access your device, I would say it doesn't really matter, you have to carry out your own risk assessment. If you shut down your device when you aren't using it, you add an extra layer of security when you put in your device password to open it as well.