I was somewhere today where a person showed me a copy of one of my posts and asked me why the copy I was being shown was lacking my avatar (which, as you can see, is me with a dinosaur behind her-I am female in case you didn't guess-, with *salon* as the screen name below). Since I"ve been putting an avatar with my posts for years, and archive.org clearly shows this going back at least through 2009/2010, I was a little stumped as to why the photo didn't show.
The techie nerd in me (I was a Microsoft network systems engineer) thought about this and I figured it out. There is a difference between a screen CAPTURE of a page as it actually exists and a screen SCRAPE done by running a mouse over a post, copying the text, and then posting into something else, like a Word document.
THIS, for example, is a screen CAPTURE. Note the design elements around the post itself, the border, the positioning of the avatar with the screen name and date underneath on the right hand side, as well as the subsidiary design elements above and to the right.

Now compare to a screen SCRAPE. I created this by running my mouse down the post, above, which did NOT pick up the photo, and then pasted into a word document. Notice how now the avatar is missing, the screen name drops below the title on the left, the border is gone, and the photo is no longer positioned on the right hand side. Also the subsidiary elements on the page are now missing. Had I taken more care to include the photo in the scrape, it would be been here as well,. I assume the person scraping the page was more concerned with the content than being visually accurate and including every design element. ANYONE that has ever dragged a mouse over content to select, copy and paste, KNOWS this and it's, frankly, disingenuous to claim otherwise.

Again, notice that if the PHOTO is not specifically grabbed, it won't show up pasted into Word.
Here is an example from 2013 that shows I have been using avatars for a long time. (Again, provable on archive.org)

And here's a post I did in 2011 making fun of Sid Miller. That's also one of my cartoons. I think I pretty much have had a border around postings here, as a design feature, for years.

And from March 16, 2010

Even back in March 2009 I had a picture with my posts, although then it was a picture of my cat and not me.

So now you REALLY know what a nerd I am.
P.S. Is it UNUSUAL for a copy and paste for a website to miss getting a picture? Nope. Here's an article describing the issues with copying and pasting a webpage where the image doesn't come over. And here's one about missing pictures in word that were copied from a webpage. AND
Copy/Paste
"In general, copy/paste is a reasonable approach when you want to save only a portion of text that you see on a web page."
OK, so if you still want to take the copy/paste route there are approaches, but there's almost no chance of getting exactly what you see in your browser. Depending on the page design and the program you're pasting into, there are many things that will not copy over or will copy over slightly differently. Consider that the same exact page viewed in two different browsers, for example Internet Explorer and FireFox, will look slightly different. You'll see the same exact page, and yet not the same exact results.
If different browsers which are specifically designed for viewing web pages can't get it the same, then the chances of other programs such as Word doing so are basically zero.