I had a problem with the smileys as well. They don't actually appear by default, but if you view your email in plain text and not HTML (as I do) there they are.
Thanks! Stylish style installed and seems to work. But it's still bewildering why Fastmail does that. Anyway, I must say it's impressive that they make an email service where this is my biggest complaint :)
I made this for Stylish to fix the issue: https://userstyles.org/styles/106482/fastmail-hide-smileys/