Try to add the following code snippet to the Custom CSS area in Theme Settings to do this

.woodmart-header-links > ul > .menu-item-register:hover > .sub-menu-dropdown, .woodmart-header-links > .menu-item-register > li > .sub-menu-dropdown{
    visibility: hidden;
   opacity: 0;
    pointer-events: none;
    animation: unset;
    -webkit-animation: unset;
}
.woodmart-header-links > ul > .menu-item-register > .sub-menu-dropdown.login-opened{
   visibility: visible;
   opacity: 1;
    pointer-events: visible;
    animation: unset;
    -webkit-animation: unset;
}

and this part to the Custom JS

jQuery('.menu-item-register a').on('click', function(e){
   e.preventDefault();
   jQuery(this).siblings('.sub-menu-dropdown').toggleClass('login-opened');
})

As for the logout so it is how WooCommerce works and it doesn’t depend on our theme.