I traveled solo on the Big Island not too long ago for 2 weeks and rented a car and never felt unsafe. It is easy to navigate around the island. I traveled the entire island and never had any worries as a woman traveling solo. In my opinion, just buy a good travel book and ditch the tours.
I personally would recommend the Mauna Kea Beach Hotel - I stayed there for 5 nights. It is pricey, but definitely worth it if you can swing it. Also, I was not a fan of the Kona area at all - very touristy. I stayed 1 night at the Kona Seaside - it was nice, but I would have preferred to have spent an extra night elsewhere.
To visit the south part of the island, you will need more than 1 day. Volcano National Park can easily take more than 1 day. It really depends what you like to do. If you're into hiking and outdoor activities, then you're going to want to spend more than 1 day.
Feel free to message me if you have any specific questions or want additional info.