Personally, I've not had the best luck with the jiffy pellets. It seems like they get too wet, then stay wet too long, then all of a sudden they are too dry and difficult to get wet again.
The white fuzzy stuff growing on the sides may not be damaging to the seedlings, but it indicates a problem with the moisture regulation. A symptom, rather than a cause.
The way your plants look while dying does suggest that they are being over watered.
You can start seeds in a plastic clamshell like you get cookies or strawberries in. There are holes in the bottom of these containers for drainage. When I use these, I line the bottom with a paper towel to keep the soil in, but allow water to drain out. Don't let the containers stand in water for more than a couple hours or so. You can keep the clamshell closed till the seeds sprout, then open them up and turn the fan on so they get good air circulation. Otherwise, they can get damp-off. They start looking weak and dark at soil level, then just fall over.
You may have to check the moisture two or three times a day until you get familiar with how fast they dry out, and how the containers look and feel when dry as opposed to wet.
It sounds like you just need some practice sprouting seeds. Don't be afraid to try again and do things a little different. Also, I usually don't have very good luck with the really cheap bargain seeds sold at Dollar stores. Even if they germinate and grow, they may be frail plants that don't produce well.
One more thing--The grow lights need to be very close to the plants while they grow. That will keep them from getting spindly. And after germination, you may have trouble trying to grow cool season crops like broccoli along side warm season plants like tomatoes. What makes on happy will make the other one mad. Kind of like trying to live with two women.