As i understand you want to edit the user avatar, this is created by the username.

First, publish or copy the file menu_user_dropdown.blade.php in you vendor theme.

To change the letter, find per example this code in tabler theme:

<span class="avatar avatar-sm rounded-circle">
            <img class="avatar avatar-sm rounded-circle bg-transparent" src="{{ backpack_avatar_url(backpack_auth()->user()) }}"
                alt="{{ backpack_auth()->user()->name }}" onerror="this.style.display='none'"
                style="margin: 0;position: absolute;left: 0;z-index: 1;">
            <span class="avatar avatar-sm rounded-circle backpack-avatar-menu-container text-center">
                {{ backpack_user()->getAttribute('name') ? mb_substr(backpack_user()->name, 0, 1, 'UTF-8') : 'A' }}
            </span>
        </span>

Update as you need.

Then find the link who have backpack_url('logout') and comment or delete the link. This way you will hide the logout.

Cheers.