No cost just be careful, do not want to become a statistic. Would rather go to botanical garden for picnics safer because its closed of from the outside.

There is street parking all around. With car guards to keep an eye on your vehicle

There is no need to book, the place is open to public except the government offices side. However, I suggest you move as group and do not carry any valuable items in the evening, My TIP- Always be observant of individuals moving up and down the parks , all day long. You find them in almost all the parks. So try to stay away from parks in the evenings

Yes it is fairly safe during daytime. Church street is a major feeder route and very busy so be careful in crossing the street. The Sheraton hotel is directly across from the Union Building on the southern side and a good one to stay at. I am sure the hotel would offer help as well. Hope you will have a splendid time.