[Vlasenko Fedor]

<style>
  .yellow {
    background-color:yellow;
  }
  .none {
    background-color:transparent;
    -webkit-transition-duration: 1.5s;
    -o-transition-duration: 1.5s;
    -moz-transition-duration: 1.5s;
    transition-duration: 1.5s;
  }
</style>
<div class="row">
  <label for="name">выделить ссылку
    <br>
    <span id="select_color">цветом?</span>
    <span class="req">*</span>
  </label>
  <select class="styled-select" id="counts" onchange="show_country(this.value);">
    <option value="0" selected disable>Нет</option>
    <option value="1">Да</option>
  </select>
</div>
<span id="q1" style="display: none">
  <div class="row">
    <label for="email">выберите цвет
      <br>для ссылки
      <span class="req">*</span>
    </label>
  </div>
</span>
<script>
  function show_country(val) {
    var el = document.getElementById('select_color'),
      timerId;
    if (val == 1) {
      el.className = 'yellow';
      timerId = setTimeout(function () {
        el.className = 'none';
      }, 30000)
    } else {
      el.className = 'none';
      clearTimeout(timerId);
    }
  }
</script>

используйте CSS свойство transition