Related symptoms?

My cat is 6 and has never had any real issues. Recently she’s been puking more often. It’s always right after eating and undigested kibble, so I assumed she eats to fast. She’s an orange cat and always hungry and eats fast. I’ve been giving her more wet food and using a slow feeder instead of her automatic feeder all the time and that helped for a bit, but she’s back to puking occasionally.

I switched her food to science diet sensitive stomach and mixed that in. She went a day or two not eating much (for her). Now she’s eating again but I just noticed she’s losing hair on a spot on her back, and there’s a red spot. Are these related? Going to vet soon.

*I wanted to add a pic of her hair spot but it’s not letting me