Theme adaptions

Actually you are asking many questions in one reply so let’s address it one by one.

Is there anyway that you have a child theme that keeps the social icons at the top right replacing the search bar,

The solution for this is here

The child theme in the thread you have linked has the sidebar on the left hand side but I would like it on the right.

You can change it from theme or page layout settings as shown in the attached screenshots.

It also like to have the social icons on the top right, and the menu and icons font to be larger. Is this possible?

Yes it is possible to achieve this by developing custom CSS code.

After doing above please share me your site URL where it is displaying so that i can help you to develop custom CSS code.