Is there a way in changing the company's logo without editing the CSS itself? So when a certain page/link is clicked, the logo changes accordingly.
You can use custom fields to change the logo per page or try this plugin - https://wordpress.org/plugins/advanced-custom-fields/ to make things easier. Just create a field named 'logo', assign it to all pages and you will now have provision to upload logo in the edit page. Use this code to display the logo
<?php the_field('logo'); ?>. That's it!