Macbook Air second screen - battery drain

No, none. I have done this for years with a multitude of Mac notebook models and never experience any negative issues doing so. The Apple power adapter will regulate the amount of charge applied to the battery and will not overcharge it past 100%.

You're very welcome! Also, that convertor, if plugged directly into one of your notebook's ports will draw power as well. As such, it's always better to be sure your notebook is connected to power when using external devices ... especially displays.

Because, your MBA is working "harder" pushing video to both its built-in display and the external display?