Ok then could you please as their support whether the add-on uses WordPress wp_nav_menu_items hook to display navigation menu which Ivory Search plugin uses to display search form in the menu?

I hope you are well today and thank you for your questions.

I would like the search bar to appear on the right-hand side within the menu bar for browser. Currently it is below the menu.

You can achieve this by adding the below CSS code in the Custom CSS option of Ivory Search plugin settings on the path Ivory Search -> Settings -> Settings

@media only screen and (min-width: 767px){
#header .is-menu form.is-search-form {
    max-width: 200px;
}

#header .is-menu form.is-search-form label {
    width: 72%;
}

#header .is-menu form.is-search-form input.is-search-input {
    max-width: 100%;
}

#header .is-menu form.is-search-form input.is-search-submit {
    padding: 7px 4px;
}
}

For mobile I would like it to apepar at the top of the nav menu above bikes. Currently it is at the bottom.

You can display it directly in the mobile header as described below.

Display Search Form In Mobile Site Header

Best regards,
Ivory Search Support

Sorry to hear about the problem you are having.

Could you please share the page URL from your site where it is happening so that I can troubleshoot it?

Best regards,
Ivory Search Support

Sorry to hear about the problem you are having.

Could you please share the page URL from your site where the search form is displaying but not working so that I can troubleshoot it?

Best regards,
Ivory Search Support

I hope you are well today and thank you for your question.

The dropdown search form style doesn’t work for shortcodes.

Would you mind if I log in to your site and do some troubleshooting? If this is ok then could you please temporarily share me your site log in details privately by checking the option “Set as private reply” when replying to this topic?

Best regards,
Ivory Search Support