No, the importer imports all of your old forum data... Thats forum names, thread names, thread posts, users, profile information, polls, sticky topics, etc.
The redirectors redirct the old links to the new locations.
I certainly hope you don't have TELNET access to a server, Telnet is insecure. It'd be liely that your host has SSH installed...
The importer doesn't need command line access to do its thing, it just needs to be run on the server... I however don't think it can be run from a seperate server, I think it has to be run wherever the old files are at...
Creating a database dump however is a nessessity if you're moving... And PHPMyAdmin isn't an option (nor is any other web script) when creating a backup of a large database, as the PHP Maximum Execution time will kill the download before you are able to download a complete backup...
If you cannot create a database dump, i highly recommend asking your host to create one for you... especially if it is scheduled to be decommissioned...