Hi Joss,
Yes, you define them in Admin and 'link' them to one, several or all categories. They do not appear before you have selected a category in sidebar of search (assuming in Admin the custom field(s) is/are 'linked' and "search" option is set to enabled). This is how the regular search sidebar is setup.
I guess you could try to create a "View" on existing custom fields, loop through them all and use them in your horizontal search bar somehow if you want them to appear from the start and disable/remove the regular hook that adds the custom fields.
As for the styling/CSS, you can use your F12 in browser and see that custom fields on search side bar all use same NAME syntax => meta + counter. There is a function in core class "\oc-includes\osclass\frm\Field.form.class.php" which contains the function "static public function meta". That function is creating the custom fields and also includes the CSS classes being used. Once you know those CSS classes you can probably "overrule" them using your own CSS class(es) OR just modify the css in core which is not recommended with regard to future Osclass updates.
SmaRTeY
I'm sorry for the delay of my answer
I would like to transform the vertical search menu of the sidebar into horizontal menu as the home page but for the custom fields they do not appear directly in the code I do not know how to modify the appearance of each of these fields
Thank you very much for your patience and help