I used to live in Vancouver, and you definitely do not need a car. The public transit is fantastic! We had a car, but we only used it for weekend jaunts out of town or to load up on groceries. You can easily use the Skytrain and buses to get around the city. In fact, the Waterfront station for the Skytrain is very close to Canada Place, which is where your cruise will embark/disembark.
There are a lot of rules about short-term rentals, and illegal ones are quickly shutdown. In our condo building, someone was running one, and it didn't last long.
When we initially visited the city for my husband's job interview, we stayed at this Holiday Inn on Broadway. My parents stayed there as well when they came up to visit us. It's a good location with easy access to the Skytrain. Also, you can walk to many places - Olympic Village, Seawall along False Creek, Granville Island.
You only have a small amount of time, so I wouldn't recommend leaving the city. Stanley Park has already been mentioned. Our favorite park was Pacific Spirit Park, which is near the UBC campus. It's heavily wooded and peaceful. We found it more relaxing than dealing with the crowds at Stanley Park. You would need to take a bus to get there. Queen Elizabeth Park is nice but much smaller. It has a rose garden, fancy restaurant, and conservatory. You can reach this via the Skytrain. Some fun neighborhoods are Commercial Drive and Main Street. Be sure to check out both the coffee and brewery scenes.