I would like to know why the connection to an object does not allow me to see ALL the Fields? I have another ORG with the same Object and it allows me to add more Fields although some are still missing.
The connection runs with the permissions of the Analytics Cloud Integration User. This means that if that user doesn't have access to any Custom Fields then they won't show in CRM Analytics. This user comes with a default set of permissions for all standard objects but this means that custom fields and objects still need permission granted to them.
As per Salesforce guidance here you should clone the standard Analytics Cloud Integration User profile, modify the cloned version and then assign it to the Integration user. This can also be achieved via a Permission Set assigned to the Integration user.