You can never be sure that an (html) email looks the same to the recipient as to you its composer. It will depend on what email app (I include desktop programs and webmail) is used to display it. Many email apps filter the content to exclude images and links that could be used to track whether you open or read the email. (Thunderbird does.) The fonts available to the composer and the recipient will differ unless they use exacly the same operating system, and that will affect text size. Some mobile email apps do not have the ability to render some html features. Display sizes vary enormously...

Most email apps will render the basics OK - heading levels, bold, italic, colours, relative (but not absolute) text size. If you stick to those it should be comprehensible to your readers even if it looks different. An email is, after all, a message not a publishing system. If you want it to look the same to everybody, send a pdf!

Thunderbird includes the Firefox browser's rendering (display) engine. What you see during composition should look similar, but not identical, to what a recipient sees if using a browser - i.e. webmail. But the 'window' size will probably differ which will change line lengths and may displace images.

Most html emails are sent in 'multipart/alternative' format. It contains both your lovingly-crafted html version, and the same in plain text - which will lack bold, italic, colours, etc. Some (rather few) email apps cannot display html so they show the plain text. Rather more recipients prefer to read emails in plain text - it can be quicker to get the gist of them. I do on small devices. So they never see the message as you composed it, and naybe you never checked what they are actually seeing ;)

If the person replying to your email chooses to reply in plain text they may dump the html part entirely and what you'll see is the plain text version of your email with quote indents '>'. Is that what you mean by "the quoted version looks horrible"?