Yes! The sites have to be connected through one Hub site. Then the calendar webpart will be able to share content between different sites. You can easily configure a hub site through the SharePoint Admin Center.
Great! You can upgrade it into a modern site by creating a communication site. Then go into the SharePoint admin center and swap site location for the two sites. The old one will be archived while the new will be your main. Note that documents and other relevant information should be migrated to the new site, before swapping.