If you keep them cool (not frozen) and dry they should keep for a year or more.
Will regular tap water work, or will I have to use distilled or rainwater?
Tap water will do in a pinch. Distilled or rain water will likely contain fewer contaminants.
Fluorescent lighting. Will it work well for staggered stage growing year-round?
For starting the seeds it will be fine. Keep it close to the seedlings or they are likely to get "leggy". Raise it as they grow. As another Freeper suggested, they will need to be transplanted outdoors in order for them to flourish. A fluorescent bulb will not provide enough lumen's to insure that lower leaves receive the light they need.
Humidity?
Too much and the plants are likely to develop fungal / bacterial problems. Too little (combined with inadequate watering) and they will wilt.
Someone told me that nicotine is a natural insect repellant. Is this true?
True. Nicotine sulfate is a poison. A crude insecticide can be made by soaking tobacco leaves in water (as one would make sun tea).
What's the best soil to use?
Any normal fortified seed starting mix will be fine. As the plants grow macro (N,P,K) and micro nutrients will be required in sufficient quantities. They can be incorporated into the soil prior to transplanting. Signs of nutrient deficiency will indicate one or more is lacking. You can probably google up pictures of this if necessary.
How long should it be before something actually grows?
If the seeds are not covered too deeply, I'd expect to see initial signs of growth in a week or less.
bump