The version available at f-droid (2.0.2) doesn't even have the two buttons at the top that you mention to display a keyboard. Gmote 2.0.2 is suppose to do it automatically when a text input is selected like the browser's address bar, but it wasn't doing it for me, which I guess is the reason the developer decided to add a keyboard button in 2.0.7.
What I ended up doing is installing "Hacker's Keyboard" available in f-droid and google play. That keyboard has an option called "Show Permanent Notification" witch will add an icon in the notification area that will allow you to make the keyboard show where ever and when ever you need it, meaning, you can use it to make the keyboard show while in Gmote.
Hopefully this work around will work for you too.
The reason I'm not using version 2.0.7 and it's not available in f-droid is that the source code was never published by the developers, and last update was made on 2012 so that probably may never happen. Hope for San I-GNU-cio that I'm wrong.