Hello,

1- We have visited the Checkout page on your Site, but we can’t seem to find the said thing for the Desktop view: https://snipboard.io/yIRHwL.jpg

The search is appearing for the Mobile view specifically, so we applied the following Custom CSS on your Site to remove it:

.woocommerce-checkout .wd-search-full-screen {
    display: none !important;
}

Location to find Custom CSS, go to Dashboard >> Theme Settings >> Custom CSS >> Custom CSS for mobile section >> there you can find the CSS pasted.

This is the resultant preview after applying CSS: https://snipboard.io/hyctdv.jpg

Please check back your Site after completely clearing the browser and server cache.

2- Regarding the quantity input out-of-date concern, on your Site under the Dashboard > WoodMart > Patcher > you haven’t applied Patch #430083 on your Site > which is related to UPDATE: WooCommerce template. So, we have applied it now and this out-of-date concern is resolved now.

Best Regards