Hi there,

1. Keep in mind that the logo link will return to the home page

to add a button or other element, open /sections/header_section.php and add 

<a class="ajax-link" data-type="page-transition" href="link to the home page">
This is the link to the slider page    
</a>

after line 25

Replace the text 'This is the link to the portfolio page ' with a fontawesome icon for example

2. try

.light-content #logo img.white-logo {
    width: 60%;
    height: 60%;
}

to change the size of the logo (change the width and/or height)

Add the code to customizer -> additional CSS