Hello,
First of all thanks for choosing our Theme, we are glad to be you in the WoodMart WordPress family :).
I saw the screenshot you attached.
1) Please try adding the following Custom CSS in the Custom CSS for Mobile area under Theme Settings >> Custom CSS.
.wd-nav-mobile li.menu-item-wishlist {
display: none;
}
.wd-nav-mobile li.menu-item-account {
display: none;
}
2) Sorry but it is the default behavior that in mobile the search button opens then the mobile side menu and you cannot display ajax search like on desktop.
Regards.
Xtemos Studios