Just to add a little more complication for you the 757 also has an emergency gear lowering system. Press a button and an independent system releases the uplocks. The gear should then drop down under its own weight and hopefully lock.
Crossing the sahara would be no problem if the a/c has the ETOPS kit - 757s cross the atlantic regularly. Otherwise just go down through Egypt, Sudan keeping within an hour's flight of Cairo, Khartoum, Addis and so forth.
At typical ldg wt the book says you need about 1450 metres - 4750 feet - of rwy for landing. In practice it will stop in about 1100 m in the dry if you really try. For a max wt takeoff at african temps I would think you'd need around 2400m.
more thoughts to follow
In the ultra marginal TO situation it is possible to get airborne and climb one or maybe two hundred feet whilst the a/c is in ground effect (think of it as a cushion of air between wing and ground) but then to stall. This has caused some crashes when TO has been made with wrong flap setting (747 at Nairobi many years ago for eg).
If you had ground dropping away beyond the end of the rwy you might be able to execute a dramatic recovery - brushing treetops etc could make useful copy.
Good luck with the novel.