Yes the place is safe. I would suggest you to take a hotel near Laxman Jhula as camps would be uncomfortable if you're alone. the Bungy place is around 20 Kms away from main city so you can either take the bus that goes from Laxman Jhula or hire an activa for around 300 rupees per day.
if bungee jumping is the only thing u seek, you should stay at zostel or similar hostels which will be very economical and will connect you to local/travelers around. Checkout packages for bungee jumping on klook.com, i found the prices cheaper than jumpin heights for jumpin heights. also they charge an additional ₹100 for entry to the park and optional ₹750 for videography of the sport. If you want to do river rafting as well. Checkout riverraftingrishikesh.com with all sorts of packages from 2N to 4/5Nts with varying inclusions.
Rishikesh is always a safe place to be at. You will get lot of buses to reach Rishikesh. Once there you will notice lot of adventure sport offering, however visit an authorized adventure sport person (they have govt approvals) before proceeding into any activity.
It's safe in day and night for women. you can travel there without worrying. you can check the packages after visited there. Don't go for online packages. they're costly. when you visit Rishikesh, talk to localist for packages. There are hundreds of Packages.