Hello,
1) If you use the contact page with the built-in template go to Theme Options > Contact Form > add the map API https://gyazo.com/7ac83e15f5573b519c074c0f45711df8
2) Could you, please, explain in more detail what box you are talking about? Maybe provide screenshot of the box. Are you talking about the content of the post?
3) Provide us with WP Dashboard access to check the home page, please. What error do you face with the carousel? Can’t you add the element? Does not element display the correct products?
4) Built-in contact page use the Contact page template https://gyazo.com/8848925971012f6ce3b010a81037060a All the available options for this page you can find in Theme Options > Contact Form. If you want custom layout then use the Contact form 7 plugin and WPBakary to create a new page with custom layout.
Regards