Hello There,
Thanks for writing in!
1.) The post titles do not show up because you have added this custom css in Appearance > Customize > Additional CSS:
header.entry-header {
display: none;
}
Please remove this code and the post titles will be resolved.
2.) Could you please send us the urls of the post that has no content in it?
3.) To remove the border, please add the following CSS code in the X > Theme Options > Global CSS (http://prntscr.com/evui3r)
.entry-featured {
padding: 0;
border: none;
}
4.) To change the icon over the image when you hover it, please add the following css code in your child theme’s style.css;
.entry-featured .entry-thumb:before {
content:"\f004";
}
You can get the CSS rule of your desired icon from this site: https://fontawesome.bootstrapcheatsheets.com/
You must add this code in your child theme’s style.css so that it will work and will not be overwritten with the theme options.
5.) Please keep in mind that you cannot edit the blog index with Cornerstone or even using the default WordPress editor. For more details, please check “2.) Create a blog page” section in this codex article: https://codex.wordpress.org/Creating_a_Static_Front_Page#Creating_a_Static_Front_Page
Hope this helps. Please let us know how it goes.