I am a little concerned about having those buttons on the homepage - I don't see any options to disable it only from the homepage.
You can control where the buttons appear by going to Settings > Sharing in your dashboard.
I am wondering whether the big files from FB, G, pinterest, etc are (eg. big .js) are downloaded once or they are downloaded for every like button?
The sharing libraries are only loaded once.
I noticed that the Pinterest button was not displayed properly though. It seems that your theme or one of your plugins loads an old version of jQuery instead of using the version that comes bundled with WordPress.
It can create multiple conflicts with WordPress and with other plugins like Jetpack. You can read more about it here:
Try switching to a different theme for a few minutes, and check if that solves the issue; if it does, you'll want to look for the function loading jQuery in your theme's header.php file, or in the functions.php file. Once you find that function, you can remove it and enqueue the proper version of jQuery as explained here:
If the issue is not related to your theme, please try to disable all your plugins, one at a time, until you find the one that causes the conflict. You can then enqueue jQuery using the method I linked to above.
You can also contact the theme or plugin author and ask them to solve the issue.