Hello @iacovoni

Align logo and telephone to the left with a margin of 20px
Align quicklinks and header image to the right with a margin of 20px

The solution for above queries would be adding custom CSS.
But it is not guaranteed the given CSS would work perfectly, you might have added other custom CSS for your site’s customization.
However you might give it a try by adding following given CSS.
Go to Admin Panel / Appearance / Customize / Theme Options / Advanced Options and paste the following CSS.

@media screen and (min-width: 1300px) and (max-width: 1600px) { 
	.site-logo-link, .custom-logo-link, #quick-contact {
		margin-left: -6%;
	}
	#masthead {
		background-position: right;
		margin-right: 2%;
	}
	.quick-links {
		margin-right: -6%;
	}
}

Also.. is there any code to use the fontawesome in other parts of the site?

About above question, your need is not quite clear could you please explain in much more detail.

Hope this will help you.
Let us know how it goes.

Regards!!!

Hello @iacovoni

Is there a way to put socials *always* on top-left and quick links *always* on top-right regardless of screen sizes?

For the solution of you issue you need to ass custom CSS but the responsive view may be effected.
With the help of CSS only the above will be the possible solution, otherwise you might need to go through code customization for which you may need to hire a customizer.
The CSS is below:

.header-social-wrapper {
    float: left;
}

Hope this will help you.

Regards!!!!