[SalesForce] Unable to Perform SOQL Query Involving Lookup Field

In the query example you provided above, "What" is already the Opportunity, you don't need to "drill-in" to a reference off of that field. So, you should be able to access Opportunity fields using What.[Opportunity-Field-Name] as long as the fields you are querying are Standard fields.

To query custom fields from a related object on Tasks or Events, you could use TYPEOF in a SOQL statement, but you'd need to contact Salesforce to get it enabled first.

I'm not entirely sure if this would work because I don't have TYPEOF enabled, but you could try this:

// where getOpportunityId is a method that returns the Opportunity Id
My_Opportunity_ID = getOpportunityId(); 

Assnmnt = [
    SELECT 
        Id, OwnerID, Owner.Name, Subject, ActivityDateTime, ActivityDate, 
        DurationInMinutes, StartDateTime, EndDateTime, Description,  WhatID, WhoID, 
        Who.Name, 
        TYPEOF What 
            WHEN Opportunity THEN Account_Address__c, Account.Name, 
                 Account_Phone__c, Customer_Contact__c 
        END
    FROM Event 
    WHERE Id = :Asn2Create 
    AND WhatId = :My_Opportunity_ID
    AND What.Type = 'Opportunity'
];

The WhoId and WhatId fields on Task are Polymorphic (meaning that they can point to multiple target objects. Unless you have the SOQL Polymorphism feature enabled. The only "related" fields that you can query are those that are on the 'Name' API object.