If you need the Live Search element wherever you will enter its shortcode.
[live_search_xforwc]
This will work in Elementor, Gutenberg, WPBakery Page Builder or even Cornerstone. Just enter the shortcode in the appropriate Shortcode element.
If you need to limit search within a category use . . .
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Are there any more of these Wigets for filter etc. Shortcodes/Elementor
Yes, use WooCommerce Products shortcodes https://docs.woocommerce.com/document/woocommerce-shortcodes/