Is it possible to minimize the filepicker modal while it's uploading?

Yes, this is possible and is done by using hide:true as part of your picker options.

This option hides the modal immediately after a file is selected in single file mode, or once the user clicks the 'Upload' button in multiple mode. The upload will continue in the background and progress callbacks will continue to be fired as the upload happens. By default this is set to false. This only works when the picker is in modal mode (container: 'modal').

Choose several files below to see an example of how the Hide function works. When the uploads are the complete the text will say Ready!

    And here is the code that makes this work

    <script>
    $('#fpButton').click(function(){
         $("#results").html('');
        filepicker.setKey("Your_API_Key");
        filepicker.pickMultiple(
            {
                services: ['COMPUTER', 'FACEBOOK', 'BOX', 'GOOGLE_DRIVE', 'DROPBOX', 'IMAGE_SEARCH'],
                // Hide the modal and handle progress in a custom way
                hide:true,
                // With backgroundUpload:false set, the upload will start only once the upload button is clicked
                backgroundUpload:false
            },
          function(Blobs){
              Blobs.forEach(addResults);
          },
          function(error){
              console.error(JSON.stringify(error));
          },
          onItemProgress
        );
     });
    
    function onItemProgress(data){
        var selector = $('#element_'+data.id);
        if (!selector.length){
            $("#results")
            .append(
                '<li id="element_'+ data.id+ '">'+
                data.filename +
                '<div id="progress_'+data.id+'"></li>'
            );
        }
        if (data.progress === 100){
            addResults(data);
        }
    }
    
    function addResults(data){
        var selector = $('#element_'+data.id);
        selector.html('');
        selector.append(
            '<a target="_blank" href="'+
            data.url + '">'+ data.filename+'</a> Ready!'
        );
    }
    </script>