Hello,
From the symptoms it looks like some involvement of eye lash hair only and not the follicles. Just eat a healthy protein rich diet and take some multivitamin supplements for some days. The most important nutrients are biotin, silica, magnesium, sulfur, zinc, and vitamin B6. There are many eyelash growth products in the market, but their efficacy has not been scientifically proven. You can apply the vitamin Eas it has good effect on hair growth.
It is very difficult to precisely confirm a diagnosis without examination and investigations and the answer is based on the medical information provided. For exact diagnosis, you are requested to consult your dermatologist. I sincerely hope that helps. Take care and please do keep me posted on how you are doing.
What a really scary event!! You're so lucky it isn't worse. If your skin suffered less than a first degree burn, the hair follicles are in good shape. Be patient, new lashes will grow. Glad you're using vit. E. There's also a new product called vitalash and vitabrow (kinda on the expensive side) that assists follicles with new hair production,should you not see the results you're hoping for.
No they won't need to fall out before new lashes grow. It will take time so try to be patient and continue applying vitamin E to heal any damage. You cannot tell by the naked eye whether your hair follicles have been damaged; you will need to see a specialist to inspect this with the proper tools.
Think of it like laser hair removal. It usually takes 2-3 treatments before hair is completely removed and even then it is not guaranteed that you will never see hair growth again. Luckily your lashes were only singed part way and therefore unlikely to be damaged at the root. You should notice improvement within a month or two.