increase the size of the "Delivery method"

Hi,

I'm not sure what "delivery method" list you're talking about.
Could you provide a URL of the page you're talking about and a screenshot of the page with the area you want to change ?

Hi,

You should be able to change that by adding such CSS in the CSS of your template :
.hikashop_checkout_shipping_block #hikashop_field_dropdown{
width : 400px !important;
}