How to show categories on homepage?

Also on the free version demo I see option to add your business:
https://dev.quantumcloud.com/phone-directory/

How do I get that button there?

I am sorry. Somehow I missed the email notification for your questions.

1. In SBD categories are not used as much as Lists are the base pillars. Only in the pro version, the Categories are used in Tabbed mode and for the multi-page mode. Examples:
https://dev.quantumcloud.com/simple-business-directory/category-tab/
https://dev.quantumcloud.com/simple-business-directory/directory/

2. You can enable that from Settings->Enable Add New Button and then for Add Button Link you can insert a full URL to a page on your website where you can add a contact form for people to submit their requests.

Hope that helps.

Thanks