Am I able to have a career in psychology despite suffering from mental illness myself?

Most likely. There are a lot of professional people with psychological issues. Certainly there maybe some influence based on the type or severity of the illness.
Interference level. It depends on how much your mental illness interferes with your ability to function. You could seek therapy and med management through training to help ease the symptoms. You could also undergo analytic training, which involves 3 or more time per week personal analysis, that helps get to root causes of what bothers you, and how to function optimally in the profession.
Probably, not always. Idividual circumstance likely determines that - depends on how well illness is managed/treated, what type of manifestations periodically show, and if there is judgment about not practicing when one should not. Professional boards protect the public safety not professional rights - one will need to always maintain the appropriate standard of care. Thaat said, even boards have programs to help.