Second, I've installed the DTS 2000 Designer from the Feature Pack. How, and from where do I access this tool?
From the Migration Wizard, I'd recommend that your destination migrated package be physical .dtsx files. This way you can open, inspect and test your newly converted packages using BIDS. Once you are happy with your SSIS package, you can deploy them to an Instance of SSIS by right-clicking on the SSIS project and choose Properties. From here, you will see the Build and Deployment Utility options in the tree view. You may also consider at this point to locking down and saving your BIDS solutions into source code control.
You can also deploy packages from with Management Studio by connecting to your Integration Services instance from Management Studio and drilling down into the File System and/or MSDB trees. Right-click on either File System or MSDB and choose Migrate. Keep in mind there that SSIS packages can be deployed to either a file system or to MSDB.
As for your last question, the DTS Designer tool allows you to open and modify your DTS packages from SQL Server 2005 Management Studio. From Management Studio, if you connect to a 2000 instance (because this is where your DTS packages reside) and choose Management → Legacy → Data Transformation Services, you will see a listing of DTS packages in that 2000 instance. From here, you can right click on a package and choose Open and you will see what looks exactly like the DTS Design environment under SQL Server 2000. That's it…that is how you access the DTS Designer. The Open choice simply won't be there unless you've installed the DTS Designer from the SQL Server 2005 Feature Pack.
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