You can do anything a facility is willing to train you to do. Many places allow you to go straight into a critical specialty unit but some require medsurg experience first. It's all up to the facility.
Yep, this is totally true. You will have to do the research, but you will find that lots of hospitals are now having "new grad internships" for a particular field in nursing.
I graduated this past December with an ADN, and I start my new grad internship in the pediatric cardiac ICU in 2 weeks. This particular hospital will train me for 1 year, classroom and bedside. if you want it, go for it!