In case you want to remove comment from home only, Then follow these steps.
Click on edit of homepage.
Click on Screen Options and the top right.
From there check Discussion.
After this below text editor you will see new option box for discussion.
Deselect Allow comments from there
Let me know if you face any issue.
If in the root of the folder of your theme there is no file "comments.php". Than comments_template () include the file from the path "/wp-includes/theme-compat/comments.php".
To make changes in that file you can copy "comments.php" from this folder to the root of your theme. And then change as you like.
To keep the date/time text, you have to do minor modification in comments.php just add call back argument in wp_list_comments function.
'callback' => 'custom_format_comment_listing'
Then add Following code in your functions.php file. you can modification other html as per your design.