Are there any risks of doing both calisthenics and weight training on the same day?

Not really: I like cross training. I see no reason to do both on the same day.

Go ahead: Remember that calisthenics and classic weight training(with dumbells, barbells, machines, etc) are both forms of resistance training, except in the case of calisthenics your body is the "weight". Calisthenics tend to involve more of a cardiovascular component but the same principles can be incorporated into classic weight training too. Just listen to your body, so that you don't overdo it!