Hello and welcome to hamster central
This "puking face" you're describing is likely just her emptying her cheek pouches. Hamsters do this once they've transported their food to their caches. However, like Velma has mentioned, a respiratory infection could be a real possibility. Especially with a swollen/runny nose and sneezing. I would recommend you book her a vet appointment soon so she can be checked out. Respiratory infections kill in small rodents
It's crucial that Bambi and Willy are separated immediately. They will fight and kill one another soon, but before that, they will breed and you could end up with 6+ more Syrians! Then the vicious cycle begins, and many novice owners become trapped. Often, being inexperienced, they're unable to properly sex the pups and thus, siblings breed causing the population to grow exponentially. This leaves you with severely inbred and unhealthy pups! Besides the mess of hamsters you can soon find yourself surrounded by, birthing is tough on mothers, and sometimes things go wrong, and you'll end up spending hundreds to help her. There's a VERY good chance she is pregnant so be prepared for pups and pregnancy. I would seek the advice of an experienced breeder for helping condition a mother for birth, and for tips on pups. Best wishes
