Database Context and referencing user tables

Is it possible to perform SQL statements referencing user tables from within a stored procedure by passing in the database as a parameter?

Context: I have multiple databases with the same table structures that require the same stored procedure executed against them.  I would like to be able to reference user tables for processing within these stored procedures.  I keep running up against "Invalid Object Name" errors.  

Example below created to run against Northwind database which creates the same error as I get in my real application.  The DB_Name() will display as 'Northwind' just prior to the SELECT * FROM customers. But the Invalid object name 'Customers' message comes right after this.  

Running a query against the system tables from within the Stored procedure brings back expected results.  I have tried to solve this using dynamic sql but keeping running into scope problems related to temp tables or cursors that are needed in the logic.



EXAMPLE:

CREATE PROCEDURE sp_DbQuery @database VARCHAR(20), @mode tinyint=0

AS
BEGIN

IF @mode = 0
BEGIN
   DECLARE @lvSQL varchar(4000)
   SET @lvSQL = 'EXEC ' + @database + '..sp_DbQuery ''' + @database + ''',1'

   PRINT 'DbQuery = ' + @lvSQL
   EXEC(@lvSQL)
END
ELSE
BEGIN
PRINT DB_NAME()
SELECT * FROM Customers INTO #TempCust
....Additional logic which uses temp tables and cursors

END

END