I would like to disable the PIN/Password of my Android screen lock but I am unable to do that as the relevant option is greyed out. Please see screenshot:
(Click image to enlarge)
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"Because you've turned on an accessibility service, your device won't use your screen lock to enhance data encryption."
You need to turn off accessibility services before you can change any settings here.
Settings-> System section-> Accessibility-> Turn off each process using the accessability services until you are allowed to change your password again.
A second cause may be what Journeyman said, but you seem to be running an older android version that doesn't support that.
It looks like your options for a non-secure lock screen are grayed out due to a security restriction. If you have, say, a work email account on your phone it probably requires to password protect the whole phone for security reasons. If this is the case, a possible solution would be to use a different mobile email client like Touchdown which only locks the app itself instead of the phone.
This can happen because an app has added a user certificate to the credentials and requires a screen lock as part of it's security policy. MITM proxy apps or VPNs that support https do this. Once the app is removed often the user certificate is left along with the restriction making the option to remove the screen lock appear disabled.
To fix this go to settings->security->credential storage then select 'clear credentials'. this will remove user certificates and security restrictions which will allow you to remove the screen lock. If you select 'trusted credentials' to display the list, the CA certs from the installed android package should be reloaded.
This worked for me on Android 4.1.2.