Re Route and visa
In 2005, we (Irish) had to get visas from Embassies in Sudan coming and Kenya going. I believe you may find it impossible at border. (Unless you pass through an unmanned border crossing).
If you want the fastest (african speed) route I would recommend the Moyale border crossing coming from Nairobi, Isiolo, marsabit.
If you want the scenic route (much slower but worth it in my opinion head up to Lake Turkana by going Nairobi, Nakuru, Lake Baringo, Maralal and on up to the Omo Valley in Ethiopia - sometimes you can "miss" the border this way. I met a few overlanders in the past who came into Kenya unchecked through this crossing and it was a small formality to go through immigration etc in Nairobi. I don't know if this is as easy in Addis Ababa. During the rainy season it can sometimes be quite difficult taking this route and as an alternative you could choose to cut across the Chalbi dessert from Lake Turkana to Marsabit which is the route we did in reverse.
Things may have changed a little since 2005 (a bit more tarmac) so I stand to be corrected.