the seedling lying flat is probably "damped off." It's a fungal condition that little seedlings are very vulnerable too, that has to do with too much moisture and too little air circulation. There's tons written here about it if you type damping off in Search the Forum keyword box. But yes, by the time they are lying flat, they are probably goners and can't be revived.
How long back did you plant the pepper seeds? What kind of peppers? Bell peppers usually sprout pretty quickly, a week or less. Some of the super hot peppers are reputed (I don't grow them) to take considerably longer.
Peppers need warm soil to germinate. If the soil is too cool and moist they will just sit there and rot. You can see what is going on by digging around in one of your pots and trying to find the seeds you planted. If you find a seed and it has a little root sprouted, planted it back and just have patience. They do put down roots before you see anything on the surface. If you find a seed and it has not sprouted, but is solid and swelled up from how it was when you planted it, give it more time. If you can't find any of your seeds, they rotted out. Start over with warmer soil. Browse in this seed starting section for how people provide heat to their seedlings.
How deep do you plant your seeds? I don't really even bury seeds, just press them firmly into contact with the soil and then sprinkle a little bit of potting soil on top to cover them. Burying too deep can slow down their sprouting.