Hi Patrick,
Thank you for contacting us.
1) and 2) Author archive page in the demo https://demo.themonic.com/io-pro/author/matt/
refers to a page when author name is clicked. I think the screenshot you have shared is of a normal post and not a author archive page.
Please create some posts under your Author and author page will automatically appear like the demo, to see it just click on the author name.
3) The color is not saved yet, in the latest version, you have the ability to select background widget color of your choice via Customizer -> v3 Control Panel -> Design Settings, please refer screenshot below:
You can change Widget Title background color as per your choice.
the title of the page disturbs me. How can I make it disappear like on your demo
For this you can use a No Title page template while creating pages.
Refer: https://themonic.com/online-documentation-usage-guide-for-iconic-one-pro/#pagetemplates