When you move to a new province, you apply for their health care card. The waiting time is then 3 months but you are covered by your old province health care until you get the new one.
I don't know which 3 years you are talking about.
In each rolling 5 year period since the day you landed, you must spend at least 2 years in Canada. Those are the PR residency requirements.
To be able to apply for Canadian citizenship, you must have spent at least 1095 days in Canada within a 4 year period. If you never left Canada during the 3 years since you landed, you can apply for citizenship right away. If you have left for vacations, you have to make it up by staying longer.