What CMS integrations are available for CiviCRM?
CiviCRM has a complete set of APIs that you can use for such purpose, but you would have to custom-develop the part that sits within this other CMS. So for example you can use (Contact, Get) with the email address of the logged-in user in the CMS to achieve function 1 in your list, or (Contact, create) to create leads in CiviCRM from a website form and achieve function 2 in your list, or (Activity, create) to track website activities in the CRM and achieve function 3 in your list.