Is it possible, from a Flash movie, to let the user download a file from the server and browse for a file locally and upload the file?
Welcome, jakob. 
You can use the [font=courier new]getURL()[/font] function to download the file. The problem might be that the file will be opened automatically if its type (such as image or audio files, for instance) is recognized by the browser.
getURL("URL", "_blank");
However, you could use a server-side script to force a download and handle the uploading process. The question is, do you have any server-side scripting languaje installed on your server? :-/
…By the way, you might find some info in the Server-Side section of the forum; do a search in there. 
