Can I change the colour of the item-count badge on my header cart icon?

Yes. Go to Settings → Appearance → Layout Settings, on the Header tab, and click Edit on the cart row under Header Components. The cart fields are:

• Show Count: Yes shows the number of items, No hides it.
• Badge: With Badge shows the count. Without Badge hides it.
• Badge Text Color and Badge Bg Color: pick your colours.

You can also change the Label, choose to show or hide it, and set an icon. Click Save.

The cart editor also shows a Display option (Link or Mini Cart). Your storefront doesn't use that setting at the moment, so changing it has no effect.