Не будет и не должно, потому как первым в списке дочерних элементов у родителя "team-descr" — это H3.
<div class="team-descr">
<h3>Bradley Hunter</h3> <!-- это :first-child -->
<p class="name">Based in Chicago. I love playing tennis and loud music</p>
<p class="name">Based in Chicago. I love playing tennis and loud music</p>
<p class="name">Based in Chicago. I love playing tennis and loud music</p>
<p class="name">Based in Chicago. I love playing tennis and loud music</p>
<p class="name">Based in Chicago. I love playing tennis and loud music</p>
<p class="name">Based in Chicago. I love playing tennis and loud music</p>
<p class="name">Based in Chicago. I love playing tennis and loud music</p>
<p class="name">Based in Chicago. I love playing tennis and loud music</p>
<p class="name">Based in Chicago. I love playing tennis and loud music</p>
<p class="name">Based in Chicago. I love playing tennis and loud music</p> <!-- это :last-child -->
</div>
Под
:first-child/:last-child
подразумевается не вхождение выбранного Вами селектора, а первая нода, тег, которые одет по списку.
В ваше случае псевдо-элементы будут работать только в том случае, если в контейнере первым и последним будут ".name"
<div class="team-descr">
<p class="name">Based in Chicago. I love playing tennis and loud music</p> <!-- :first-cild -->
<p class="name">Based in Chicago. I love playing tennis and loud music</p>
<p class="name">Based in Chicago. I love playing tennis and loud music</p> <!-- :last-child -->
</div>