How to increase the width of the sidebar menu

If you are using the first predefined header then, you can use the following code in your template custom CSS:

body.predefined-header1 .body-wrapper {
    width: calc(100% - 16%);
    margin-left: auto;
}

body.predefined-header1 #sp-header {
    width: 16%;
}

 Templates > Styles > Click on your current default template name then choose Template Options > Custom Code > Custom CSS

How to change paddings inside the sidebar menu

For that task also Custom CSS is needed:

body.predefined-header1 #sp-header {
    padding: 40px 20px 30px 20px; }

you can control the sidebar inside space by changing the value of paddings, the default values are 60px 35px 30px 35px - in my example, I used smaller values.

How to decrease the spacing between the logo and menu

For that task, you have to use Custom CSS. In the below example, there is also a CSS override to reduce the spacing between menu items. Default values (from template styles) are  130px and 33px, and as you can see I reduced those values 3 x times:

  @media screen and (min-height: 550px) {
body.predefined-header1 #sp-header .sp-megamenu-wrapper {margin-top: 50px;}
body.predefined-header1 #sp-header .sp-megamenu-wrapper .sp-megamenu-parent > li,
body.predefined-header1 #sp-header .sp-megamenu-wrapper .sp-megamenu-parent > li:not(:last-child) 
{margin-bottom: 10px;}
}