Just to clear things up, this is called "User Flair", and exists on the StackExchange network data for a user under the 'flair' tab, or accessible by specific URLs such as the site-specific one:
... which only shows the reputation on that specific site, or the StackExchange-wide one:
(Both of these are mine, by the way)
User Flair such as that is not customizeable beyond the theme in use according to all the Meta.SE threads I checked for this.
Known themes to exist are (appended by
?theme= to the PNG URL):
- None (the default)
dark - makes a dark theme
clean - can't tell the difference from default but it works
hotdog - makes a lot of red.
I don't think you can customize this to have anything such as "latest answer" or "most voted questions" - at least, there isn't any such 'flair' that I have found for this. I went hunting for such things and didn't find them. If they exist, they aren't documented, though I highly doubt there's any type of 'custom' embed-able code that gets you the other stats.
(There may have been at one point, that was done via a third party service, but not anymore.)
And just to confirm I made inquiries via certain discreet channels, and got a "Yeah there's nothing other than user rep you can put there" as the reply.