Thanks, Jupp.
The Player Client and the GM Client can both display the map full screen. If you have two laptops networked together with an ethernet cable (or wireless), one for the GM and one for the players (hooked up to a projector), you'll have some sweet capabilities at your disposal in a face-to-face game session. What won't be possible in this initial release is to output both the GM and the Player's views from a single laptop/desktop.
The initial release will let you "paint" and erase only on the Fog of War layer, but in a future version I am planning to add a separate Drawing Layer (including text labeling tools) with which you'll be able to improvise quick maps (hopefully with cool texture fills, too). I will see if I can include tablet support, though you're the first person to mention this need.
I'd recommend using a specialized map-making tool like Dundjinni to create for yourself a series of reusable locations to have a small library at your disposal whenever you need to improvise an encounter. Those maps can be used as a starting point on which you can add furniture, trees, etc., to make the encounter feel more unique/customized. No one likes playing on the exact same map over and over.
I also hope and expect that the "User Creations" section of the Battlegrounds forums will serve as a good central repository of stock maps, figures, and props that people will upload to and share with the rest of the Battlegrounds community.
As for spell effects, you should know that you will have a wide selection of semi-transparent overlays in different sizes available for use. These arcs and circles will be provided both in standard and gridded mode (where the arc or radius of the overlay is already translated into discrete grid squares or hexes).
I hope this answers all your questions, and I encourage you to post other questions directly on the Battlegrounds forums (saves me from having to answer the same questions repeatedly on various sites).
