Hi there,
You raise a bunch of questions in your post that could take a lot of space to answer...which is probally why no-one has rushed in with replies just yet, however here goes with a short version:
The most common hunting destinations include South Africa, Namibia, Zimbabwe, Zambia, Botswana, Mozambique and Tanzania.
Each of the above countries apply a different set of hunting regulations according to species available and current game population levels. This may change annually. Game is categorised as either "Plains Game" which includes most antelope species, zebra and giraffe or as "Dangerous Game" which would as the name suggests traditionally include dangerous species like lion, leopard, buffalo, rhino, and elephant.
As to importation of firearms and limitations, each country has it's own set of rules by example South Africa allows handgun hunting while Botswana does not allow the importation of any handguns.
You are correct in your view that bolt actions are most common, I've seen a few lever actions and even a slide action in use in South Africa, however blot actions are by far the most used.
Our African Hunters Guild's website (members section) contains the answers to just about all the questions you ask ...you can opt to join up as an associate members for free and find some real useful info.
Hope this helps some.