Dropdown terminal binds this key '`' (the key above the tab key; backtick) with its launch. Whenever I press this key, it launches dropdown terminal or closes it. How can I use this key to input the assigned character (backtick) in a text or in the terminal?
You are talking about the Gnome extension, I assume? That extension uses "the key above the tab key" by default, whichever character that may correspond to in your specific keyboard layout.
According to this issue in the extension's issue tracker, it doesn't provide a way to type the character that's "blocked" by the shortcut. But you can change the shortcut key in the extension's settings, for example to CTRL-`.