Tailwind

Unfortunately, there are no Fitness facilities at this hotel.  If it is a hotel near Haneda Airport, "Hotel Mystays Haneda" is not big, but there is also a fitness gym and it is free for guests to use.

The hotel is right inside the airport, so u don't need to walk out

There is.

There are 2 plan,with breakfast plan and room charge only. If you can move, its directly connected to the terminal building, so there are plenty of cheap and deliciousfood to eat at the terminal building restaurant !

Since it is directly connected to the airport, it becomes an airport parking lot.

Nearest should be Shinagawa station. But Shinkansen stops there just for a minute. I would prefer Tokyo, neighboring station, because it is the terminal. You have enough time to get on a train there. According to JapanTaxi app, the estimate fare is 5,500 to 6,100JPY. Uber is 7,600JPY. (I haven't used Uber here in Japan though.)

I believe check in time ends at 2:00. When I stayed, my flight came in at 22:30 and was able to check in.

I don't think so. They stored my luggage for the day in a pile next to reception with other guests luggage. I checked out in the morning and didn't fly until late at night so i went to Tokyo for the day. So stored for around 12h.