I've driven through Zimbabwe, South Africa, Kenya and Tanzania - with safari companies and by simply renting a vehicle. Road-wsie South Africa is fine, just like any european country, others are occasionally spotty, roads are no fantastic, but acceptable. We actually rented a pick up truck in Zimbabwe, which worked fine for getting around.
If you're going to do it yourself just be careful and use common sense.
As for border crossings, you cannot legally take rental cars across borders.
Been stopped by cops / army, again, use common sense. More than likely they are looking for a bribe. Best way to do it, have a $20 or something tucked into your passport or under your drivers license, when you hand it over for them to look at, the bill "accidentally" falls out on to the pavement, say something like oops and leave it up to them to pick up, leave it up to them to either pick up and return to you, or for them to pocket.