Newbie,
The "grow light" is different from regular incandescent light. It is more like the sun's light. It's not especially for bonsai, but for any indoor grown plant or plants that don't receive enough natural sunlight. You may not need it, but you had said that the neighbor's balcony overhangs and cuts down on your light, so it could help take the place of the natural sunlight. Also, remember that bonsai just like any other plant, drinks through it's roots. If the watering isn't sufficient enough to drip through the drain holes in the bottom of the pot after you have just watered, then it's most likely not getting enough water. Just be careful to not over water. Like Jose suggested, put a stick on the soil to check for moisture. Kind of like checking to see if a cake is done by poking a toothpick into it's center to see if it comes out clean. As for the window being open, I don't think that would hurt anything.
Also...IF by some chance this little elm doesn't make it - any of us who have gotten interested in bonsai, have all had our share of misfortunes. It's how you learn.
