Dear uchika96 ,
Before touch the hardware you need clear understand that the fault can't be fixed on software level.
Try to flash using Odin tool : fallow the instructions
if the firmware flashing gets failed try again to flash it
as you mentioned snow rain effect comes due to fault of EMMC, if you have sound BGA reworking skills you can replace the chip
[http://www.ebay.com/itm/Emmc-Memory-For-Samsung-S3-i9300-KMVTU000LM-B503-/321327750136?hash=item4ad0a05bf8|Here] you can buy item if required.
i did EMMC replacement on 2014 December and phone still works fine. so many professional volunteers here will help you to fix your phone. let us know the result