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Problems are solved. My only problem is that I don’t know how to translate the theme to another language. Please help me if you can.

For the solution of multiple search boxes and overlapping sidebar problem, look below:

Tinkering around with it a bit more, I realized that I may be able to solve the problem with a few changes on site-content css class. Some attributes that singlepost has and site-content doesn’t have causes the sidebar work on sigle posts flawlessly and on the homepage with problems when the screen has a lower width than 768px (style.css, line 1402).

I’ve found the problem.
1. I removed style="width:100%;" from the line
<div id="col" style="width:100%;" class="column">
completely instead of changing it.
2. When I first tried to add the sidebar to the homepage, I added the sidebar on the top of the php file under the header: this was a mistake. I added it at the bottom of the page, before the footer, and made the necessary changes in style.css after that. So the bottom of home.php reads:

<?php get_sidebar(); ?>
<?php get_footer(); ?>

3. I added
.site-content{width:72%;float:left}
before the line
.singlepost{width:72%;float:left}
(line 1397) so that the homepage content will cover 72% of the screen on high-res screens and

4. I added
.site-content{width:100%;float:none;}
before the line
.singlepost{width:100%;float:none;}
(line 1402) so that the homepage will cover all of the screen on low-res screens and behave exactly as the singlepost pages (widget area after the content)