Install SharePoint Designer on your machine and access any site that you have permissions for. SPD does not live on a server.
Re the list: in the list settings click "Save list as template", then in the Site Collection go to the Site Actions > Site Settings > Solutions. Download the solution file. Browse to the other server's site, upload to that site collection's solution gallery and activate the solution. Now you can create a list from that list template on the other server's sites.