There are several things to notice. In short, it is not possible to configure this via CSS, nor are some of the requested features possible.

But one by one:

1. padding
This actually works. And this is shown in the examples. As it is not working in your case, I assume you specified the tree-structure-painter for a label (e.g. TREE_COLUMN_CELL). But that is not working. The tree-structure-painter CSS property is only available on table level. This is also documented in the N&N of 1.4.

public class TreeExample extends AbstractE4NatExamplePart {
    @PostConstruct
    public void postConstruct(Composite parent) {
        natTable.setData("org.eclipse.e4.ui.css.CssClassName", "modern");

/* MODERN CONFIGURATION */

.modern {
	tree-structure-painter: background padding tree;
	table-border-color: auto;
	cell-background-color: white;
	text-align: left;
	padding: 0 5;
}

.modern {
	tree-structure-painter: background padding tree;
	table-border-color: auto;
	cell-background-color: white;
	text-align: left;
	row-height: auto; /* I added this line */
	padding: 10 10 10 10; /* I increased the values */
}

row-height: auto;
padding: 5 10 10 10; 

The tree-structure-painter CSS property is only available on table level. This is also documented in the N&N of 1.4.

2. alignment
This also works. But maybe not as you expect. The tree state icons are always rendered left aligned. But the content can be aligned differently. I assume the issue is the same as above.

3. decoration icon left to the tree status
First, your assumption is wrong that the supported values for tree-structure-painter are only background, padding and tree. In theory you can add also other available painters like decorator. That would be the solution, but currently the TreeLayer doesn't respect a CellPainterDecorator while searching for the IndentedTreeImagePainter. For that I created a ticket and proposed a fix, so it is possible to create such a painter programmatically. But it won't work on CSS level as the tree-structure-painter needs to be specified on table level, the decorator value expects the decoration property, otherwise it fails to create it, and you definitely don't want the decoration on all columns.

https://bugs.eclipse.org/bugs/show_bug.cgi?id=498920

There are several things to notice. In short, it is not possible to configure this via CSS, nor are some of the requested features possible.

But one by one:

1. padding
This actually works. And this is shown in the examples. As it is not working in your case, I assume you specified the tree-structure-painter for a label (e.g. TREE_COLUMN_CELL). But that is not working. The tree-structure-painter CSS property is only available on table level. This is also documented in the N&N of 1.4.

2. alignment
This also works. But maybe not as you expect. The tree state icons are always rendered left aligned. But the content can be aligned differently. I assume the issue is the same as above.

3. decoration icon left to the tree status
First, your assumption is wrong that the supported values for tree-structure-painter are only background, padding and tree. In theory you can add also other available painters like decorator. That would be the solution, but currently the TreeLayer doesn't respect a CellPainterDecorator while searching for the IndentedTreeImagePainter. For that I created a ticket and proposed a fix, so it is possible to create such a painter programmatically. But it won't work on CSS level as the tree-structure-painter needs to be specified on table level, the decorator value expects the decoration property, otherwise it fails to create it, and you definitely don't want the decoration on all columns.

https://bugs.eclipse.org/bugs/show_bug.cgi?id=498920

org.eclipse.nebula.widgets.nattable.examples.e4.part.TreeExample.postConstruct(Composite)
        // turn the auto configuration off as we want to add our header menu
        // configuration
        final NatTable natTable = new NatTable(container, gridLayer);

org.eclipse.nebula.widgets.nattable.examples.e4.part.TreeExample.postConstruct(Composite)
        // turn the auto configuration off as we want to add our header menu
        // configuration
        final NatTable natTable = new NatTable(container, gridLayer, false);
        ....
        ....
        natTable.addConfiguration(new DefaultNatTableStyleConfiguration());
        natTable.configure();

2. alignment
This also works. But maybe not as you expect. The tree state icons are always rendered left aligned. But the content can be aligned differently. I assume the issue is the same as above.