pagination showing all entries
When you use {% paginate %}, the second variable that's returned, pageEntries, is  the paginated entries. When you do craft.entries.section('news').featured('not 1') later in the template, you're doing a new query that's not paginated.

Assuming what you want is to return all featured entries at the top, then a paginated list of news below, try something like this:

{% extends "_layout" %}
{% set title = "News" %}

{% paginate craft.entries.section('news').featured('not 1').limit(5) as pageInfo, pageEntries %}

{% block content %}
    &lt;section class="news-page section-wrap news-list"&gt;
        &lt;h1&gt;Company News&lt;/h1&gt;
        {% for featuredEntry in craft.entries.section('news').featured('1') %}
            &lt;article class="news-list--item news-list--feature"&gt;
                &lt;div class="feature-img"&gt;
                    {% for image in featuredEntry.featuredImage %}
                        &lt;img src="{{ image.getUrl('featuredArticleImage') }}" alt="{{ featuredEntry.title }}"/&gt;
                    {% endfor %}
                &lt;/div&gt;
                &lt;div class="feature-content"&gt;
                    &lt;h1&gt;&lt;a href="{{ featuredEntry.url }}"&gt;{{ featuredEntry.title }}&lt;/a&gt;&lt;/h1&gt;
                    {{ featuredEntry.body.getPage(1) }}
                    &lt;a class="btn btn-secondary" href="{{ featuredEntry.url }}"&gt;Continue reading&lt;/a&gt;
                &lt;/div&gt;
            &lt;/article&gt;
        {% endfor %}

        {% for regularEntry in pageEntries %}
            &lt;article class="news-list--item"&gt;
                &lt;a href="{{ regularEntry.url }}"&gt;
                    &lt;h2&gt;{{ regularEntry.title }}&lt;/h2&gt;
                    {{ regularEntry.body|hacksaw(words='25', allow='&lt;p&gt;&lt;b&gt;') }}
                    &lt;span class="cta-border"&gt;Continue reading&lt;/span&gt;
                &lt;/a&gt;
            &lt;/article&gt;
        {% endfor %}

    &lt;/section&gt;
    &lt;nav&gt;
        {% if pageInfo.prevUrl %}&lt;a href="{{ pageInfo.prevUrl }}"&gt;Previous Page&lt;/a&gt;{% endif %}
        {% if pageInfo.nextUrl %}&lt;a href="{{ pageInfo.nextUrl }}"&gt;Next Page&lt;/a&gt;{% endif %}
    &lt;/nav&gt;
{% endblock %}


If additionally, you only want to show the featured entry on the first page, you could wrap the first for loop in:

{% if pageInfo.currentPage == 1 %}
    ...
{% endif %}


More details on pagination in the docs.