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

1) I tried with the default style, and now the button and the box are in two lines. I tried with some css code you wrote in the forum but can’t work it out. Can you please help me?

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

.is-menu form label {
    max-width: 80%;
    display: inline-block;
}
.is-menu form input.search-submit {
    width: auto;
    margin: 0;
} 

2) it will be great to have it in the right side with a narrower button. also, I see other people have the small “search icon/symbol” button rather than the text. How can i change it?

You can achieve it by changing menu search form style other than Default as described on the below page.

Change Menu Search Form Style

3) I also tried some other styles, but then I can not see anything display in the menu area, which does exist when I move the mouse pointer to the position. is it because of the color of the menu is black? what I can do if I want to use slide or other style?

Yes, you are right as the search icon has black color.

You can change seach icon color to white using below custom CSS code.

.is-menu path.search-icon-path {
    fill: #FFF;
}

Best regards,
Ivory Search Support

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

You will see those options and display the search form in the menu as described on the below pages.

Display Search Form In Menu

Change Menu Search Form Style

Also, could you please share your site URL where it’s displaying so that I can help you?

Best regards,
Ivory Search Support

I hope you are well today and thanks for posting here.

I will implement it in the next version of the plugin which will be released soon.

Best regards,
Ivory Search Support

I understood what you are trying to achieve.

Could you please tell me how you have created the footer on your website and share the screenshot of the footer from your website admin area where you have created it?

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

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

.elementor-inner .is-form-id-13993 label {
    width: 100%;
}

Best regards,
Ivory Search Support