If the car-hire companies have offices in Botswana and South Africa, then you can usually take cars into these countries. You will need to advise the company in advance, as they need a few days to apply for the right permits and insurances – which may cost an extra N$150 or so.
If you want to do a one-way hire, this is also possible, but expect a one-way drop-off fee of around N$1,000–1,500. Note that car hire is generally cheaper in South Africa, and about the same price in Botswana. Thus for a long trip a one-way hire from South Africa into Namibia is usually slightly cheaper than vice-versa.
Taking vehicles across Zimbabwe's borders is trickier, and has only become possible in the last few years. It is generally very expensive to do one-way hires that pick up or drop off in Zimbabwe (though Botswana's Kasane is very close).