I suspect there are a few problems here. I suspect they're too wet, and the ones that "leaned right over" have "damped off," which is caused by a fungal problem that usually comes from too much moisture, like Masher suggested. I also suspect that maybe the ones that came up "without leaves" might be so-called helmetheads? This occurs when the seed's husk stays on and, as a result, the cotyldons ( that initial set of two weird leaves) cannot escape. I've heard this can be caused by burying the seeds too shallow, but care must be taken to not overcompensate... Bury too deep, and they'll be starved for light before they even hook and break out of the soil. 6-12mm is probably the sweet spot.
Like Masher mentioned, you'll probably want to provide some artificial light for the plants. That may or may not be a contributing factor to the problems you're already experiencing, but it will surely lead to problems in the future. Some other pointers that i'd like to share that may or may not be a factor here:
-I used to start seeds in paper towels, but found that if the taproot grew more than maybe 1-2mm long on the paper towels, they'd start to grow into the towel and it became impossible to extricate them without damaging the new and delicate roots. I've since moved to paper coffee filters, and that is no longer a concern. As a side note, it seems like most experienced and confident growers put the seeds directly in the starting media, but I'm too anxious for that. I need to see that little taproot pop out so I know the seed is viable...
-which leads me to my next point. If you're starting on paper towels or coffee filters, you want to put the seeds in the starting media as soon as the tap root pops out. Aside from the aforementioned risk of root damage, I have also found that seeds that developed in the filters for too long tend to struggle alot in these crucial first few weeks. Legginess is a concern, as is root development and overall health/strength.
-I use that Jiffy starting media. I'm sure there are better choices, but it has always worked well for me, and the other 2 seed-starting soil products I've used have lead to disaster; I'm scared to try anything else. Bottom line, I can vouch for that stuff. Might not be the best, but it's completely adequate so if you're wondering if it's a problematic factor or not, I wouldn't worry about it.
In closing, yeah, like Skullbiker said, pics are GREAT for troubleshooting these kinds of problems. It allows THPers to see right away if you got helmet heads, if sprouts are damping off, if they look leggy, even if the media looks too dry or (more likely) too wet, etc. Even if you're not experiencing problems, THP loves pics so every thread is better with photos.
Hth!!!