Quora (ie. small round avatar next to username

Please navigate to the Dashboard  > Comments > Settings > Styles admin page put the following CSS code in the "Custom CSS Code" textrarea:

#wpcomm .wc-comment .wc-comment-left {
   width: 33px !important;
   margin-left: 58px;
   margin-top: 16px;
}

#wpcomm .wc-comment-label span {
   display: none !important;
}

#wpcomm .avatar {
   border-radius: 50%;
}
#wpcomm .wc-blog-post_author > .wc-comment-left .wc-comment-label {
   display: none;
}

just username without avatar

Use the following code:

#wpcomm .wc-comment .wc-comment-left {
display: none;
}

In any case please don't forget to delete all caches and press CTRL+F5(twice) on the frontend before checking.

1) How can I customise user avatars to look like either Reddit (ie. just username without avatar) or Quora (ie. small round avatar next to username)? I'd prefer to go for the Quora stlye..

You just need to use the following CSS code. Please navigate to Dashboard > Comments > Settings > Styling admin page, put the following CSS code in the "Custom CSS code" textarea:

#wpcomm .wc_social_plugin_wrapper {
    display: none;
}

Please don't forget to delete all caches and press Ctrl+F5 (twice) on the frontend before checking to reload the updated CSS file.    

Please follow the Astghik's answer in this support topic: https://gvectors.com/forum/wpdiscuz-widgets/avatar-appears-to-be-extremely-large-before-the-comment-section-on-the-post/#post-21923

1) How can I customise user avatars to look like either Reddit (ie. just username without avatar) or Quora (ie. small round avatar next to username)? (Attached image: massive avatar.png).

2) When I log to my website, there's a huge avatar icon below the phrase "You are logged in as User | Log out". How can I remove this? (Attached image: avatar customisation.png).