Add image to sidebar?

That space is not editable, it is your background image.

If you want to do this you will have to design your background image in advance and upload it appropraitely depnding upon your template.

You can try to make a background image and align center.
The image corner at both end can be what you want(i.e width greater than the header width).

You can try to make a background image and align center.
The image corner at both end can be what you want(i.e width greater than the header width).

How can I ensure that the body part will stretch with longer body lengths?

I checked and found your website container width is

.container {
width: 950px;
}

So to make the far left and right to display the repeating image, you can try any background image width greater than 950px with center alignment background.