Amazed... if the plants themselves look healthy, don't worry about the mold/whatever on the soil. Those little "mini greenhouses" are horrible... there just isn't enough air circulation in there.
In general, I only plant 2-3 seeds in each "compartment" and then, when transplanting (although I plant in much larger planters, so they don't need transplanting early), I don't worry about trying to save each plant. If two of them are far enough apart that they can be separated without breaking off or damaging the roots, great-... I get two plants. If not... I transplant the whole soil block, and snip off the weaker plants, leaving one.
The best way to transplant is to carefully "pop out" the entire soil block- usually pressing up on the bottom of the container will do it. DON'T pull on the plants, or if you absolutely have to, don't use the stems... handle them by the leaves. The stems contain their "blood vessels", so to speak... if you crush them, you kill the plant. If you break off a leaf, providing there's at least a little bit of at least one leaf left, the plant will probably survive just fine.
I don't use peat pots (is that what you mean by "brown cuppy things"? LOL!) because they dry out SO easily. And, while you can transplant the entire peat pot into the garden, if you don't bury every bit of it, the part which is above ground will "wick" the moisture out and often the plant dies. They also can sort of "trap" the roots.
If you want "cheap", use something like dixie cups... punch a few holes in the bottom, put them in a tray of some sort (cheap aluminum foil baking pans work well). Use a good potting soil (or "soiless mix"). Put an inch or so in the bottom, (depending on how much soil comes with the root ball when you move the seedling). Place the seedling in the cup, gently holding it at the level you want it, and carefully scoop soil in until it's completely filled around the roots. You don't want air pockets... gently firm it around (press down around the roots with your fingertips). Water well (this is why you put holes around the bottom of the cup... so excess water drains out) and put them in bright light or sun.
A fan blowing on them will help prevent damping off (from any bad fungus you may have in that "green stuff")
If you have several seedlings in a single container, and you want to save as many as possible, then you need to use something- a paring knife, or small metal spatula can work, or a baby spoon- to gently scoop them out of the soil. In that case, fill your Dixie cups or whatever to within 1/2" of the top with potting soil, and use your finger to make a hole in the middle, about as deep as your root systems are long. GENTLY dangle the bare roots of the seedling into the hole (you want them to go straight down if possible, not be curled up) and then press the soil around it.
Give them a couple of days before you worry overly much about how they look, although young seedlings generally don't show much transplant shock.
Don't fertilize if you're using a good commercial potting soil... most of them contain enough fertilizer to last the seedlings at least 3-4 weeks. After that, you can use a DILUTE solution of Miracle Gro or one of the organic versions.
If I missed something, or confused you, let me know!
Summerthyme