I was in 4 African countries over a 5-ish week period starting in Sept 2001 [not hunting], well before the more recent strife that you might be talking about. More recently, there was a push by governments/ruling factions to reclaim land/farms from the white settlers/owners and that is where a lot of the stories you hear about come from. There are definitely regions where this was severe and others where it was minimal. When there, we flew into and used Nairobi as a sort of base - it was slightly sketchy in some areas; rural areas were not an issue except for the kids begging on the tourists. We only went south as far as Vic Falls and Bulawayo so not into Joberg or SA.
Similar to any international travel, you should be aware and knowledgeable. Check the state department for travel information. As noted above, with reputable guides/outfitters, most all of the logistics will be handled and you will be personally handled from the time you land at your arrival airport to the time your are returned to your departure airport. That will include customs as well. Talk it through with your contact and outfitter. Also, be very fluid, tolerant and patient when traveling there - especially across borders.
As for the hunting: from I have read - again, I have not yet personally hunted Africa - the biggest cost for hunting safaris these days is actually returning with your target species. The handling, dip&pack, shipping process is not only doubling or exceeding travel costs but is also taking extensive periods of time - even with species of non-concern/cites issues. Be mindful, ask a lot of questions and explore all the options you have available to handle this whether using a service or doing it yourself. Make sure you understand the trophy fee system. Also, make certain you understand the hunt concession - many in South Africa in particular - are large game ranches.
Finally - and I mention this with trepidation and a grain of salt only - try exploring the Accurate Reloading forums; they can be a trainwreck but if you just peruse and search there are tidbits of info to be found as the members there are well versed in Africa and African hunting. There are many outfitters and booking agents that you might be able to reach out to in your research.
Hope this helps - we had a great time; the only real problem I have is being able to get back there to enjoy it again!