input type=date then datepicker Icon is not appearing on Firefox

The input UI depends on the browser more info.

You could implement a custom design (so it will always appear the same in all browsers) using jQuery. Like here.

Firefox doesn't do that but, Here is how I would go about it:

The input outside the div is for other browsers. The one inside the div is for firefox. I use JS to detect the browser and act accordingly, hiding each.

Note that you will need to download a calendar icon and for your img src.

<!DOCTYPE html>
<html lang="en">
<head>
    <meta charset="UTF-8">
    <meta name="viewport" content="width=device-width, initial-scale=1.0">
    <title>Document</title>
</head>

<body>
    <div class="firefox" style="border: 1px solid #DDD; width:max-content;">

        <input style="border: none;" type="date" id="start" name="trip-start" value="2018-07-22" min="2018-01-01"
            max="2018-12-31">
        <img src="calendar.png" style="width:15px; padding:10px 15px 0px 0px" />
    </div>

    <input class="anotherBrowser" style="border: none;" type="date" id="start" name="trip-start" value="2018-07-22"
        min="2018-01-01" max="2018-12-31">


    <script>
        // Detect Firefox 1.0+
        var isFirefox = typeof InstallTrigger !== 'undefined';

        //Hide 
        if (isFirefox) {

            document.querySelector(".anotherBrowser").style.display = "none";

        } else {
            document.querySelector(".firefox").style.display = "none";
        }
    </script>

</body>

</html>

I'm able to resolve datepicker icon disappear in input textbox when we use input type="date". It works in Chrome, Firefox, Edge. Below are the css I have used and applied those on divs

 .cal-wrp {
    position: relative;
    background-color: white;
    margin-bottom: 16px;
 }

.btnCal-wrp {
    display: inline-block;
    width: 100%;
    position: relative;
    z-index: 10;
}

.btnCal-click {
    border: 0.0625rem solid #cfd0d3;
    background-color: transparent;
    width: 100%;
    padding: 0.75rem;
    cursor: pointer;
    border-radius: 0;
    font-size: 0.875rem;
    line-height: 1.4;
}

    .btnCal-click::-webkit-calendar-picker-indicator {
        right: -10px;
        position: absolute;
        width: 78px;
        height: 40px;
        color: rgba(204, 204, 204, 0);
        opacity: 0;
    }

    .btnCal-click::-webkit-inner-spin-button {
        -webkit-appearance: none;
    }

    .btnCal-click::-webkit-clear-button {
        margin-right: 75px;
    }

.btnCal {
    background: transparent;
    border: none;
    color: black;
    padding: 13px;
    position: absolute;
    right: 0;
    top: 0;
}

Calendar textbox code:

<div class="cal-wrp">
   <div class="btnCal-wrp">
        <input class="btnCal-click" type="date" required>
   </div>
   <button class="btnCal"><span class="fa fa-calendar"></span></button>
</div>