Ok, I asked my husband to go after the flies with his macro lens, so here is what they look like. Can anyone identify these critters? There are hundreds of them crawling all over the container!

Fungus gnats look similar. as to plants dieing, its cilantro and they do yellow after a few days on older leaves. It appears that your watering too much and that cna also deal to damping off and the base of the stems will dry out casing the plant to die. Avoid using common potting or garden soils as they can contain both issues. Sterial seed starting medium will give better results and has no fungus or insect cultures in it to cause these damages.

To remove the fungus gnats, use yellow sticky traps placed horizontally on the edges of the pots. For the eggs and maggpots they produce, use a form of Bt that will kill the maggots in the soil.

Numerous times these same issues have been asked covered here and I give the same answers repeatedly.

Here is a link that might be useful: Fungus gnat photos