However it's looking very strange.  I would like my Recent Comments to look exactly like your Recent Replies you have to the right here on this screen. 

The Recent Replies widget you've mentioned belongs to the wpForo forum plugin. You've purchased the wpDiscuz Widget add-on.  The Recent Comments of the wpDiscuz Widget add-on doesn't have the same styles (look) as the Recent Replies widget of the wpForo plugin. 

I don't want a border around my widget.  Also I would like a way to make the avatar a bit smaller.

Below are provided CSS codes you can use for this purpose: 

#widget-comments-container .wpdiscuz-widgets-content{
   border: none !important;
}
#widget-comments-container .wpdiscuz-widget-popular-comment-author .popular-comment-author-avatar-box .commenter-avatar-box img{
    width:60%;
}

As you will see the article name and name of the commenter is not being shortened

This is a small conflict with your active theme. The CSS codes below should solve the issues:

.popular-comment-author-body {
   font-size: 14px !important;
   width: 100% !important;
}

.popular-comment-author-body a {
   display: block;
   width: 83%;
   color: #566c7d !important;
}

.popular-comment-author-body div:first-of-type {
   font-size: 12px;
}

All CSS  codes should be added in the Dashboard > Comments > Setting > Addons > Widgets > "Widget custom CSS" textarea.

You can change the red mark value as you want.

I installed the Widget addon.  All I want from this widget is the Recent Comments. 

However it's looking very strange.  I would like my Recent Comments to look exactly like your Recent Replies you have to the right here on this screen.  I don't want a border around my widget.  Also I would like a way to make the avatar a bit smaller.

As you will see the article name and name of the commenter is not being shortened.  Having said that I really would like my Recent Comments to look like yours.  

Can you please help me with this.

Please see example here: https://www.discerningtheworld.com/  Scroll right down to footer.

Thank you so much for your help!