Answer: Based on your description, I am not certain what is happening; it might be a cultural conditions issue or possibly a disease issue. There may be a combination of factors at work, so I'll try to do a brief review and hope you can trouble shoot from there. Tomatoes will need about 16 hours of light a day with the light down close to the foliage, within an inch or two. The soil mix should be a soil-less seed starter mix or a good quality soil-less potting mix. This helps reduce the chance of disease and also helps with over or under watering issues.
The soil should be kept slightly damp like a wrung out sponge, not sopping wet and not dried out. There should be good air circulation in the area where they are grown. The seedlings should be thinned so they do not touch eachother, and preferably to one per container, as soon as they are large enough to distinguish which ones are most vigorous. All of your tools and containers should be washed clean before planting, and if used previously they should be soaked in a bleach and water solution as well.
Dying edges can indicate underwatering, or possibly overfertilizing, or possibly a micronutrient issue, but with young seedlings I would not expect to see fertility issues so early. It could also indicate excessively dry air or that the light bulbs are too close and giving off heat.... The spots and yellowing could be overwatering or could be a disease problem. Many tomatoes are subject to diseases that can be brought in on tools, in potting mix, or even by a person who smokes or chews tobacco. The best defense on that is to plant varieties with good disease resistance which is indicated by the letters such as V,F,N,T and so listed on the seed packet.
I'm sorry I can't be more specific for you long distance but I hope this helps.
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