It seems the image didn't attach to the first message. It's attached here. The issue also seems to occur in text that is not contained in remote content. I just received a plain text email with the issue.

Plain text messages cannot show colours or different fonts. I find it hard to believe that Thunderbird can somehow recognise text parts of images and then choose to render them improperly. So I think this is all about HTML-formatted email.

Do images and other graphics render properly?

I'd check what fonts have been selected and whether or not it is set to override the fonts declared in messages.

Tools|Options|Display|Formatting->Advanced

Okay, duplicated the issue that I thought was related to an image. It's not actually image related - not surprising for exactly the reason you suggest. Thunderbird is not somehow parsing images. When I loaded the remote content, it shuffled the layout of the email and it appeared as though the text was a part of the image.

In any case, its definitely a text formatting issue. If I view the emails in simple HTML or as plain text, the text looks normal. The original HTML seems to be causing problems. If I uncheck the option allowing messages to use other fonts, the issue goes away. Could Thunderbird be misinterpreting fonts somehow? (The emails look normal in other clients.)

The font simply might not be installed. The sender specifies realyfancefont and it is in the font family script, it is more than possible it is rendering in a completely different script font once received

You would need to look in the source of the email (ctrl+U) at the fonts defined and see if you have them installed. That is one of the reasons the web is make up of images, fonts do not translate machine to machine well.

Exactly what I was just doing. All the fonts I can find referenced in the HTML are Helvetica and Arial. No fancypants fonts to be found.

I could be missing something, of course. The HTML ain't exactly formatted nicely.

I do not know if it is just me, but I have no Helvetica font installed in Windows 10.

Nope google indicates it is not just me. So, are the fonts list family style eg

font-family: "Times New Roman", Times, serif;

Or is it a case of an orphan and the operating system makes it's own decision as to what is the "next best" member of the family.

Helvetica is the same font family as Arial, so it should just get displayed as Arial on Windows machines (that don't have Helvetica installed). Arial is correctly included with Helvetica in the font-family tags. But its moot - I have Helvetica installed.