Anacharis is a good one. It just thrives on the nutrition in the water table. It grows so easily that it's sometimes considered a weed.
The lighting in the tank definitely will have a huge impact on how the the plants perform. Without adequate lighting, no, they won't really help a whole lot the the process. Though for what it's worth, the light in my 10G Tetra Starter Kit was adequate for growing plants in my aquarium.
At the time, I had an Amazon Sword (good starter, very hardy), Green Myrio, Moneywort, Narrow Leaf Subulata, Moss Ball and I used no CO2 or ferts. Obviously, not everyone will have the same experience.
One thing you'll want to take into consideration, is not only type of bulb, and raw light output (lumens), but also color temperature. Daylight bulbs generally have the widest color range, but warmer color ranges (6,500K and up) seem to do best for plants.