In VSCode I can only deploy changes from my project to the scratch org. Is it possible to do the same when VSCode is connected to the standard sandbox?
Yes, you can use
sfdx force:source:deployand pass the username or alias of the environment to deploy to using the
Yes, it can be done. Its called as Org Development Model. Sandbox and Prod dont support dx format. They support metadata format. You can create a project with manifest and use VS code to deploy in sanboxes.
Or, convert sfdx to metadata format and then deploy.