To remove/allow toggling of UI elements around Twitch's chat, we've created this application, which you can install in your browser.

A few things worth mentioning:

Chat elements will be displayed only if Twitch chat is open. You can select this option in the settings menu. 

When Twitch chat is visible, you can hide/show the message input using the settings menu. This can be a quick and easy way to avoid chat messages getting annoying.

You can use Alt + Up Arrow, Alt + Down Arrow, Ctrl + Up Arrow, and Ctrl + Down Arrow keyboard shortcuts to slide the chat message input up and down.

And in case you don't want to use the keyboard shortcuts, you can use the settings menu to save the chat message input as a setting to automatically show/hide the message input.

When the chat message input is hidden, you'll be able to type directly in it.

And with the built-in settings, you can save this setting as well.