Notice the binary size of the app is only 64KiB.

In the demoscene this would be a disqualification as it's 676 bytes over the limit, but in this case I'll overlook it because of the sheer awesomeness of what you've done (and I'm sure trimming off 676 bytes wouldn't be too difficult ;-)

More seriously, this is an excellent proof-of-concept that a Slack client does not need to use hundreds of MB of RAM and consume most of a modern CPU core to provide its basic functionality. I'm a long-time Win32 programmer who started in the tail-end of the Win16 days and one of the things I've wanted to write if I had the time and need was a Win32 native Slack client, to show that it can be done with far less resources, but you've gone even further with Win16. Win32 has native TLS support (via SChannel library, not well-documented but examples exist) and you'd be able to even use it practically (32-bit applications will run on all current versions of Windows, both 32 and 64-bit.)

Note that C89 only prevents declarations of variables in the middle of a scope; you can simply create an inner scope with new variable declarations at its start. Like so...

    case WM_PAINT: {
      PAINTSTRUCT ps;
      HDC hdc = BeginPaint(hwnd, &ps);
      ...
    }

Another small tip: if you make it a dialog-based application, you won't have to bother with WM_PAINT and drawing text yourself --- you can just make the "statusText" and "settingsText" edit controls, and use SetWindowText (or SetDlgItemText) to set their content. They will paint themselves automatically just like the listboxes that you used for the chat contents and userlist. At least you'll save some of those 676 bytes that way...