In both Dragonfall and Hong Kong:

When I try to start a Rigger, the Drone Tray is below the bottom of the screen. Of course this means I can't use the drone.

Is there a way to correct this?

You might need to be at a different resolution. Also setting the resoltion with tv menu might cause that as well. what is resolution of your monitor/tv?

This must be specific to your configuration, since I just played a game starting with a rigger and had no such problem.

So, whatever graphical set-up you have needs tweaking. (I can personally vouch for the stability and back-compatibility of the Shadowrun engine/s, since I play on a laptop with integrated graphics, meaning that I must play with a resolution of 640×480, rather than the monitor's native resolution of 1200×768, to prevent acute narcolepsy mid-game. ;)

Try experimenting with the resolution (last tab of the in-game menu, where you can save and load). Does the problem persist with other resolutions? If so, you have some problem displaying a full screen on the monitor you have.

Also remember that any resolution changes may be laggy, so it is best to restart the game after selecting a screen resolution that is suitable.