How to eliminate the "Edit" word that appear in the search.

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

When I put mouse over the search box, appear a “edit” text below the box.
How can I delete that?

It is only displayed to the logged-in site administrator users so that that they can easily edit the search form.

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

.is-link-container {
    display: none !important;
}

Another question is about the color of submit button now on mouse out is BLack, and on mouse over is Orange, I wanna make inverse of this.

You can change the color by editing the search form and using the Search Form Customizer from the customize section of search form.

Best regards,
Ivory Search Support