Check out this helpful demo:

---

<form action="/action_page.php">
            <input class="form-box" type="text" name="search">
            <a href="#"><img src="https://police.un.org/sites/default/files/radio-microphone.png" id="google-mic" alt="microphone"></a>
        </form>

---

    form{
    position:relative;
}
input{
    width:100%;
    height: 50px;
    border-radius: 10px;
    margin:0;
}
a{
    margin:0;
    padding: 10px 5px;
    position:relative;
    right:0;
    top:0;
    position:absolute;
}
a img{
    width: 30px;
    margin:2px 10px;
}

---

Check out the demo here for more details.

Here are some suggestions to improve your styles, you may want to consider replacing your current style.css file with the following:

* {
    margin: 0;
    padding: 0;
    box-sizing: border-box;
}

body {
    font-family: Arial, Helvetica, sans-serif;
    font-size: 13px;
    height: 100vh;
}

li {
    list-style-type: none;
}

li a {
    text-decoration: none;
    color: #000;
    opacity: 0.75;
}

li a:hover {
    text-decoration: underline;
}

.header-container {
    width: 100%;
    height: 60px;
}

.nav-list ul {
    display: flex;
    flex-direction: row;
    align-items: center;
}

.nav-list ul li {
    display: flex;
    justify-content: space-between;
}

.left-half {
    margin-right: auto;
    padding-left: 15px;
}

.left-half li a{
    margin: 15px 8.5px 0 10px;
}

.right-half {
    margin-left: auto;
}

.dot-box {
    padding: 0 3px;
}

.dots {
    height: 16px;
    width: 16px;
    opacity: .7;
    margin-top: 5px;
}

.dots:hover {
    opacity: 1;
}

.sign-in-btn {
    padding-right: 22px;
}

#sign-in-button {
     color: #fff;
     border: 1px solid #4285f4;
     font-weight: bold;
     background:#4387fd;
     line-height: 28px;
     padding: 0 12px;
     border-radius: 2px;
     opacity: 1;
     text-decoration: none;
}

/* #02. Title / Search Bar */

.google-logo-header {
    display: inline-block;
    text-indent: -9999999px;
}

#google-logo {
    display:inline-block;
    width: 370px;
    z-index: 0;
    opacity: .91;
}

.main-container {
    text-align: center;
    position: relative;
}

.form-box {
    box-shadow: 0 2px 2px 0 rgba(0,0,0,0.16), 
    0 0 0 1px rgba(0,0,0,0.08);
    width: 584px;
    height: 44px;
    border-radius: 2px;
    border: none;
    outline: none;
    position: relative;

    font-size: medium;
    padding-left: 16px;
}

.form-box:hover {
    box-shadow: 0 3px 8px 0 rgba(0,0,0,0.2), 
    0 0 0 1px rgba(0,0,0,0.08);
}

#google-mic {
    position: relative;
    right: 38px;
    height: 20px;
}

.buttons {
    background-color: #f2f2f2;
    border: 1px solid #f2f2f2;
    border-radius: 2px;
    color: #757575;
    cursor: pointer;
    font-size: 13px;
    font-weight: bold;
    margin: 11px 4px;
    min-width: 54px;
    padding: 9px 16px;
    text-align: center;
}

.buttons:hover {
    border: 1px solid #c6c6c6;
    box-shadow: 0 1px 1px rgba(0,0,0,0.1);
    color: #000;
}

.btn-one {
    position: relative;

}
input{

}
.btn-two {
    position: relative;

}

/* #03. Footer */

.footer-container {
    background-color: #f2f2f2;
    border-top: 1px solid #e4e4e4;
    width: 100%;
    bottom: 0;
    position: fixed;
    line-height: 40px;
}

.footer-list {
    display: flex;
    align-items: center;
}

.footer-list li a {
    padding-left: 27px;
}

.settings-pad {
    padding-right: 27px;
}

.footer-right {
    margin-left: auto;
}

Your primary challenge in creating a responsive website is the heavy use of position: absolute; and fixed left and right positions. This approach can hinder responsiveness. The ideal solution involves utilizing Flexbox or CSS Grid along with Media Queries, all achievable with standard CSS.

If you'd like to delve deeper, here are some resources to explore:

Flexbox:

• W3Schools

• CSS-Tricks

CSS Grid:

• W3Schools
• CSS-Tricks

Media Queries:

• Responsive Web Design - W3Schools
• CSS Media Queries Examples - W3Schools

Modifications to Your Code

The changes I made to your code involve transitioning it to Flexbox for better responsiveness while maintaining the HTML structure intact. Here's an overview:

<section class="main-container">
    <img src="images/google-logo.svg" alt="google-logo" id="google-logo">
    <form class="search-bar" action="/action_page.php">
        <input class="form-box" type="text" name="search">
        <a href="#"><img src="images/google-mic.png" id="google-mic" alt="microphone"></a>
    </form>
    <div class="btn-row">
        <button class="btn-one buttons">Google Search</button>
        <button class="btn-two buttons">I'm Feeling Lucky</button>
    </div>
</section>

To enhance responsiveness, I changed the styles for .main-container to a flexbox column aligning items vertically centered:

.main-container {
  display: flex;
  flex-direction: column;
  align-items: center
}

Removed position: absolute and fixed positioning rules, ensuring elements are more adaptable and responsive. Added classes like .search-bar as flexbox row for vertical alignment:

.search-bar {
  display: flex;
  flex-direction: row;
  align-items: center;
}

#google-mic {
  height: 25px;
  margin-left: -30px;
}

Introduced .btn-row div for button alignment in a flexbox row fashion for improved responsiveness:

.btn-row {
  display: flex;
  flex-direction: row;
}

Take a look at this JSFiddle for a functional demonstration. Keep an eye out for fixed pixel widths on elements that could cause issues on mobile screens. We recommend utilizing percentage widths and refining SVG placement to minimize whitespace around the Google logo. Adjustments may be needed based on screen sizes.

We suggest exploring the provided resources and adopting responsive design practices for optimal site performance.