What is my X11 doing with Ctrl-Period and how can I disable it?

This looks like the "preedit area" of some X Input Method provider, allowing you to compose a character before it gets sent to the actual input of the application.

(For example, a preedit dialog is used when composing Japanese text in fcitx or IBus, or when entering Unicode codepoints in GTK's Ctrl+Shift+U. Usually it is the application toolkit such as GTK or Xlib which communicates with the IM module, telling it about text fields being in focus, etc.)

In this case it's likely used for emoji input by name, guessing from the 'e' prefix and the keyboard shortcut. I'm guessing that this specific "input method" is part of the IBus input framework, where emoji input is built-in and the default shortcut is indeed Ctrl+period. (Normally it should show a graphical chooser but in some cases only basic IM support is available and all you get is the preedit field.)

However, check the XMODIFIERS and GTK_IM_MODULE environment variables for additional clues. (IBus is not the only XIM server out there.)

IBus has the keyboard shortcut customizable through ibus-setup, or via GSettings:

gsettings get org.freedesktop.ibus.panel.emoji hotkey

This shows that the default value is ['<Control>period']. To set it to an empty list, use:

gsettings set org.freedesktop.ibus.panel.emoji hotkey "@as []"