Identify why a user can drop a database
I would suggest running this query on the master 

SELECT  
    [UserName] = CASE princ.[type] 
                    WHEN 'S' THEN princ.[name]
                    WHEN 'U' THEN ulogin.[name] COLLATE Latin1_General_CI_AI
                 END,
    [UserType] = CASE princ.[type]
                    WHEN 'S' THEN 'SQL User'
                    WHEN 'U' THEN 'Windows User'
                 END,  
    [DatabaseUserName] = princ.[name],       
    [Role] = null,      
    [PermissionType] = perm.[permission_name],       
    [PermissionState] = perm.[state_desc],       
    [ObjectType] = obj.type_desc,--perm.[class_desc],       
    [ObjectName] = OBJECT_NAME(perm.major_id),
    [ColumnName] = col.[name]
FROM    
    --database user
    sys.database_principals princ  
LEFT JOIN
    --Login accounts
    sys.login_token ulogin on princ.[sid] = ulogin.[sid]
LEFT JOIN        
    --Permissions
    sys.database_permissions perm ON perm.[grantee_principal_id] = princ.[principal_id]
LEFT JOIN
    --Table columns
    sys.columns col ON col.[object_id] = perm.major_id 
                    AND col.[column_id] = perm.[minor_id]
LEFT JOIN
    sys.objects obj ON perm.[major_id] = obj.[object_id]
WHERE 
    princ.[type] in ('S','U')
UNION
--List all access provisioned to a sql user or windows user/group through a database or application role
SELECT  
    [UserName] = CASE memberprinc.[type] 
                    WHEN 'S' THEN memberprinc.[name]
                    WHEN 'U' THEN ulogin.[name] COLLATE Latin1_General_CI_AI
                 END,
    [UserType] = CASE memberprinc.[type]
                    WHEN 'S' THEN 'SQL User'
                    WHEN 'U' THEN 'Windows User'
                 END, 
    [DatabaseUserName] = memberprinc.[name],   
    [Role] = roleprinc.[name],      
    [PermissionType] = perm.[permission_name],       
    [PermissionState] = perm.[state_desc],       
    [ObjectType] = obj.type_desc,--perm.[class_desc],   
    [ObjectName] = OBJECT_NAME(perm.major_id),
    [ColumnName] = col.[name]
FROM    
    --Role/member associations
    sys.database_role_members members
JOIN
    --Roles
    sys.database_principals roleprinc ON roleprinc.[principal_id] = members.[role_principal_id]
JOIN
    --Role members (database users)
    sys.database_principals memberprinc ON memberprinc.[principal_id] = members.[member_principal_id]
LEFT JOIN
    --Login accounts
    sys.login_token ulogin on memberprinc.[sid] = ulogin.[sid]
LEFT JOIN        
    --Permissions
    sys.database_permissions perm ON perm.[grantee_principal_id] = roleprinc.[principal_id]
LEFT JOIN
    --Table columns
    sys.columns col on col.[object_id] = perm.major_id 
                    AND col.[column_id] = perm.[minor_id]
LEFT JOIN
    sys.objects obj ON perm.[major_id] = obj.[object_id]
UNION
--List all access provisioned to the public role, which everyone gets by default
SELECT  
    [UserName] = '{All Users}',
    [UserType] = '{All Users}', 
    [DatabaseUserName] = '{All Users}',       
    [Role] = roleprinc.[name],      
    [PermissionType] = perm.[permission_name],       
    [PermissionState] = perm.[state_desc],       
    [ObjectType] = obj.type_desc,--perm.[class_desc],  
    [ObjectName] = OBJECT_NAME(perm.major_id),
    [ColumnName] = col.[name]
FROM    
    --Roles
    sys.database_principals roleprinc
LEFT JOIN        
    --Role permissions
    sys.database_permissions perm ON perm.[grantee_principal_id] = roleprinc.[principal_id]
LEFT JOIN
    --Table columns
    sys.columns col on col.[object_id] = perm.major_id 
                    AND col.[column_id] = perm.[minor_id]                   
JOIN 
    --All objects   
    sys.objects obj ON obj.[object_id] = perm.[major_id]
WHERE
    --Only roles
    roleprinc.[type] = 'R' AND
    --Only public role
    roleprinc.[name] = 'public' AND
    --Only objects of ours, not the MS objects
    obj.is_ms_shipped = 0
ORDER BY
    princ.[Name],
    OBJECT_NAME(perm.major_id),
    col.[name],
    perm.[permission_name],
    perm.[state_desc],
    obj.type_desc--perm.[class_desc] 


This should give you a pretty good idea of the roles that have access to your master database and see weather or not the user has any of those roles. You can also run this against any of your other databases to check the user's permissions on a database by database level. This should be an important tool to help track this down.