Check with the admissions office at the school and/or the nursing department at the school. Most schools have a list of criteria and requirements. They should be able to give a more detailed answer about specifics to the application process. Where I went to college - (BSN program), about half our class was direct entry to the nursing program that included additional criteria above and beyond the regular admissions criteria with stipulations regarding classes and grades the first year (non-nursing). We then had a very informal "application" to submit mid-spring of the year we wished to begin nursing classes (because we were all already admitted to the school, they had access to everything they needed...). Based on the information you are given, you then plan to have everything completed and submitted as soon as possible relative to when they begin accepting applications.
I would definitely type any documents that can be typed - even the people with the best handwriting have days things are not as legible as possible. So yes, type. Anything you have to submit needs to be checked for grammar, spelling, punctuation and formatting (I've had some issues with things not getting formatted correctly when I attempt to upload them online or to print depending...).
Best wishes and good luck!