RE: 'Google Apps integration enabled?'
I can confirm $Organization.GoogleAppsDomain is available in custom formula fields. So you can create one of these on the User object and do this.
User me = [select GoogleAppsDomain__c from User where Id = :UserInfo.getUserId()];
System.debug('Google App Domain: ' + me.GoogleAppsDomain__c);
RE: 'the user-specific Gmail-to-Salesforce BCC address?'
This information is held in EmailServicesFunction and EmailServicesAddress objects.
Allowing you to do these queries to discover if the service is active and what the BCC address is.
System.debug('Is active? ' +
[select IsActive from EmailServicesFunction
where FunctionName = 'EmailToSalesforce'].IsActive);
[select EmailDomainName from EmailServicesAddress
where RunAsUserId = :UserInfo.getUserId()].EmailDomainName);
The above should give you the BCC address, something like this...
Hope this answers your question! :)