Search Input and Magnifying Glass On Separate Lines

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

1. I have tried a few different CSS lines to make the search magnifying button show up on the same line as the input field but nothing has worked. Can you let me know how to fix this?

Please remove the below custom CSS code that you are using on your website.

.is-form-style.is-form-style-3 input.is-search-input {
    max-width: 90% !important;
    display: inline-block !important;
}

Then please add the below CSS code in the Custom CSS option of Ivory Search plugin settings on the path Admin Area -> Ivory Search -> Settings -> Settings

body .is-form-style.is-form-style-3 label {
    width: calc(100% - 110px)!important;
}

2. How can I get the search field to stay sticky in the header in both mobile and desktop view?

Please try using the below CSS code.

body > form.is-search-form {
    position: fixed;
    top: 0;
    left: 0;
    width: 100%;
    z-index: 9999;
}

body {
    margin-top: 36px;
}

Best regards,
Ivory Search Support