tomaja

When editing specific page, go to layout options (on the top of the right bar on edit screen) and choose “full witdh” if you dont want to see sidebars on that page.

then you can control it with

.carousel-caption {
/*your code here*/
}

For example, if you want it totally transparent you can set

.carousel-caption {
background:none; /* add color instead none if you want*/
}

The one with h1 is slider title, so if you want to remove him just use e.g.

.carousel-caption h1 {
display: none;
}

If you want to change its appearance then change font-size and color values to fit your needs. Same goes for the rest. The one with the p is for slider text bellow slider title and last one is for button (the one with .btn). If you want to change button link then you should use e.g.

.carousel-caption a.btn {
	float: right; /*moves button to the right*/
	color: red; /*color of the button text*/
}

Check this slider snippets, you should be able to do what you want. In the Change the font/size/color of the Slider Titles rules you can add display: none; if you wish to remove them from display.

It seems to me that you have changed default Customizr php and css files which is not good. I suggest to reinstall all default Customizr files. All changes should be in Customizr child theme folder.
If something is broken in WordPress as you said then you should do the same with it. Default Customizr install works good, and with logo in place where you want it.
Before you do anything, backup your site (database and all files) in case something goes wrong or you want to save css changes.

It is possible, you can use something like this for specific posts/pages e.g.

/* for specific post */
.postid-33 #customizr-slider.carousel .item {
height:         230px;
min-height:     230px;
line-height:    230px;
}

/* for specific page */
.page-id-12 #customizr-slider.carousel .item {
height:         230px;
min-height:     230px;
line-height:    230px;
}

To determine what is specific id for post or/and page use page source view option in your browser.
If you wish to use it on all posts and pages than just .page or .single should be enough

There is no need for snippet, you just need to point WordPress to send posts to your newly created page. You can do that at yoursite.com/wp-admin/customize.php , tab “Front Page”, for option “Front page displays” select A static page, and then for option “Post page” your new page.

Try with

.site-description:before {
content: '\1F4DE';
}

for adding content – icons before tagline take a look at this thread.