Hi Dean,

I started a similar tour in Cape Town last October and am currently in Khartoum so I can give you some info on the visas up to Sudan. All these were obtained with a German passport.
In South Africa you don't need a visa for a stay of up to 90 days, Lesotho and Swaziland are also easy to enter in case you want to go there. I got my visa for Mozambique in Germany but I heard that it is possible to get it at the border. Tanzania and Kenia are easy ones that you get on arrival.
To get the Ethiopian visa in Nairobi you need a letter of introduction/cover letter from your own embassy. As of Feb 2018 the German embassy in Nairobi issued those letters for 25€ but you have to book an appointment first. I heard other embassies refuse to issue those letters. With that letter you can get a 30 or 90 day visa at the Ethiopian embassy which starts on the day you get it in Nairobi.
The Sudanese embassy in Nairobi also issues 2 month tourist visas to non-residents if you bring them a letter from your embassy. Otherwise it should be possible to get a two weeks transit visa for the Sudan in Addis.

For egypt you will definitely need a carnet and it makes other border crossings easier too. Insurance is sold at the borders and you could consider getting the COMESA yellow card that is valid from Tanzania up to Sudan although it might be cheaper for you to just get the local ones.

Hope that helps you with your planning.

Cheers
Nico