I was a little confused by the comment "flying out of Helsinki at 6am". The Ferry does not return to Helsinki until after 6am arrival.
Yes, you can go anywhere within the St Petersburg port zone which is the area of the city and including the surrounding countryside,. It does not extend to Moscow or other large cities however.
You can stay in any hotel who will host you. Many have clerks who do not know about the visa waiver program for ferry passengers so will insist you need a visa. Ask to speak to the manager or just move on to another. Technically you can't stay with a host which is not authorized to host foreign guests. They should have a MBT number posted somewhere on their web site listing it as indication they are permitted to host foreigners. That is a carry over from the Soviet period when only Intourist Hotels could host foreign guests. Now, the main distinction is the size of the bond they need to post. Foreign visitor hosting requires less bond than hosting Russian or outgoing Russian guests.
The foregoing means that officially you are not supposed to use AirBnB or other non-commercial hosts. In practice however it means nothing, stay where you want. No one is going to care or check.
St Petersburg is a remarkable easy city to visit all parts of despite the language barrier. The Metro is very useful and essential if you want to be on time anywhere. But think of it as a city regional conveyance. Going from one district to another it is the best and fastest method, far better than taxi or private car. The further you go out from the city center, the distance between metro stations increases. In the city center they are more concentrated so are useful just going from one close in neighborhood to another. Also, the system of change over stations, where two lines intersect are all in the center, so it is easy to change lines underground. They are easy to get the hang of. But for closer in movement to say one neighborhood to another, the extensive bus, tram and trolley system goes where to want, from and to metro stations. The main street, Nevsky Prospect is very crowded on the sidewalks because most visitors stay in that local space but going 1 block in either direction from Nevsky and the sidewalks become free of traffic.
Reviews are so personal, they are almost useless in finding the best choice. There are over 10,000 restaurants and an equal number of cafes and fast food. You will never be further than 100 meters from a good restaurant. I have been to maybe 1000 of them and more open every day with a greater and greater focus on food inventiveness and quality. Picking any is less of a matter of luck than it was 10 years ago. It is hard to find a bad meal now. A cheap place with fresh ingredients is a chain that opened a couple years ago and has become very popular with locals is Market Place, with several of them on Nevsky Prospect. For a bit higher scale, any of the many dozens of unique restaurants that are part of the Ginza Project is a no brainer, stylish decors, each one very different. There is a specialty restaurant in the group offering just about any cuisine you would want. St Petersburg has always been an international food city so it is easier to fine Italian or Japanese than good Russian cuisine. The first commercial public restaurants from the early 1700s were Italian and French.
Have a great visit, if you have any questions just post...