which method do you recommend adding add to cart button in live search results
Those suggestions seem to come from your theme. Nothing WPSOLR can act upon.
i am trying to make category names searchable in the header search
It depends what you’re looking for:
a) Displaying products in search from a category name
Yes, this is a feature of WPSOLR. Just verify that you have selected “Index custom fields and categories” in screen 2.2
b) Displaying categories in search among other results
This is not possible, as in WordPress only searcheable post types can be displayed in archives. Categories are not post types.
c) Displaying categories in suggestions among other results
This is possible with WPSOLR suggestions, which you are not using currently.