Cairo is a huge city, with an enormous number of cars, carts, motorcycles, pedestrians, and animals. If you choose to try to drive here, you're a braver man than I, Gunga Din. It was nerve-wracking just being a passenger!
There are modern traffic signals all over Cairo now, but they appear to be mostly ignored -- in favor of a first-come-first-served model of driving...if your car fits into the space, take it before someone else does. Often, four or more cars drive abreast on a road with two lanes. It's a bit of a free-for-all. We saw very few cars that weren't banged up in Cairo. We also saw some truly horrific accidents on the desert road leading to Minya.
While it's certainly possible to rent a car or 4WD, unless you have steady nerves and no fear, I'd recommend taking advantage of the many taxis and busses in town instead.
Egyptian drivers drive on the right (or they're supposed to), and driving requires both a license and insurance (although many people drive without either). The "standard" rules of the road are supposed to apply. All front-seat passengers are required to wear seat belts.
The reality is a bit more chaotic -- lane markers are routinely ignored, stop signs are cause for only a slightl slowing, traffic signals are ignored (often whether a traffic cop is directly traffic or not), and horns are used as a kind of shorthand communication: "I'm coming through!" "Watch out!" "Get out of the way!" and seems a constant cacophany. Animal carts, both camels and donkeys, are common on even the major roads and cannot be counted on to travel the right direction. Pedestrians are a nuisance and while few are run over in the streets, it's because the pedestrians are swift and agile, not because the cars do much to avoid them!
At night, many drivers drive without their headlights on, believing that it "drains the battery", only flicking them on when they approach another car. Cars entering the road from the side streets often do not stop.
Surprisingly, most of the accidents are minor -- dented bumpers or scraped paint -- because the traffic moves so slowly.
In the city, most roads are in good condition, and the newly-created freeways throughout Egypt seem to be reasonably well maintained. In the smaller towns outside of the Delta, many roads are dirt, gravel, or (more likely) mud. In some cases, a four-wheel drive vehicle is a much, especially in the desert -- a standard car doesn't do well in the soft sand.
Most of the remote roads have periodic road blocks by the tourist police or the army, where you must stop and be logged.