I'm trying to edit my Client Profile form, but it seems the system does not let me do that from the Forms menu.

Because there is vital information in the Client Profile form that is required in order for a person to even be a contact/client in your system (name and email address), this form is always active. 

Then when you click "Edit" from the Options drop down list, you'll see the various fields within the Client Profile form. Because some of these fields are required (and considered good business practices to have in your database), they too are grayed out (name, address, phone, email, time zone, etc). These basic fields will always be included in your client profile form. It is the other fields listed here that you can choose to include or not and you can also customize additional fields.