the cotyledons (original seedling leaves) are supposed to die off so that is fine. those are very small containers even for little plants like that. I personally HATE those peat pots. When you do transplant them...
#1 get a better soil please, that stuff looks like shit. definitely not enough drainage so add some pearlite if you keep using that soil mix. I bought some Pro Mix shit from menards for some of my vegetable garden stuff... it was utter shit. after almost a month after sprouting a couple plants in the crap soil, my plants were no bigger than what you have in the picture. I suspected the soil was crap, and I removed the soil from the pot, removed as much soil from the thin stunted roots as possible, and planted in the soil mix i use in my indoor garden. 3 weeks later i have beautiful healthy plants and you would never know that they almost got tossed had they not been my last two seeds of that tomato strain.
#2 tear away those peat pots, do not leave them around the roots, and then transplant them into a container no bigger than 1 gallon asap. if you plant into to big of a container and water it down, the roots are only in a small zone when transplanted to a new pot, and can not dry the pot out fast enough. This will continue to stunt the plants growth and the roots can rot and kill the plant when kept wet for too long. Let them grow in the 1 gallons for at least 3 weeks and no more than 5 before transplanting again. this should allow you to build a healty root ball provided they are being cared for properly. Then you plant into a container the size NECESSARY for growth. 1 gal of soil = 1 month of growth. so if you go from one gallon container to a 5 gallon container your plant should be good for 3 months minimum in that container. If you transplant them in more than what they need, you are wasting resources (water, nutes, soil, etc....). Also when you transplant only water the ROOT ZONE. don't know where the root zone is? hint: it should be no more than the size of the container you just pulled it out of
So what you can do is water the plant BEFORE you pull it from the 1 gal container (which will make it come out easier anyway) and then just water it in lightly when you tuck it into the new pot. Note: if you use too big of a container, the plant cannot drink the liquid fast enough, roots sit wet, suffocate, and die (especially if you don't add more pearlite to that mix!) Not to mention when you are not watering because soil is sitting wet for too long CEC (cation exchange capacity is not effective because the roots have eaten the available nutrients around them and have nothing left to eat until you water again and get more food into SOLUTION around the roots. if you dont know what cation exchange capacity is i suggest you look into it as that is key for nutrient uptake process. The perfect soil is 50% medium, 25% water and 25% air (soil pores).
#3 As for the bugs, get some Neem oil and dilute 1/2 tsp into each gallon of water with a small amount of soap (dr. bronners is best) to help the neem coat the leaves.
Good luck with the babies! this will definitely get you on the right track. for the love of god, get some new soil lol