Hi Paul,

Welcome to Cairo.

1) What is the best method for finding housing and should we live in New Cairo or downtown?

Depends on if you want to be in the throws of Cairo (downtown) or would like to have a bit of fresh air and teh option to go into town as and when you like (New cairo). There are many areas of new cairo, Al Rehab/Al Shrouk/Kattamaya, but the one I would recommend is Al rehab, it has everything, even a bus service into town/local market area/2(small) shopping malls, a club/park with a couple of swimming pools etc.. and not far from the AUC. Al Rehab has plenty real estate agents that could show you propertys, and i also know thw AUC rents a total apartmnet block here outr for AUC teachers/professors, so the campus may be able to recommend something.

2) Is it possible to find work without a proper visa?

It is possible but difficult. Generally most expats teach english at peoples homes or do extra in a school (no work visa). This can pay between 15LE-50 LE an hour per person. As for working for companies etc... you may need a work visa which is difficult to get.

3)How much should we anticipate in paying in rent? Internet? Food?
Rent in Al rehab, can start from 2000 le a month up to 7-9000 per month depending on the size you rent and if its furnished or unfurnished.
Downtown, is around the same, but you can get chearper areas say for 1500 le a month.
Internet (wirless at home I have Dlink which i pay 1800le for a year) but you can get these wireless USB sticks from vodaphone/mobinil, that I beleive are around 500LE one off charge then a small bill for usage.
Food - if you eat local shop at a souk, small supermarket etc..for 2 you may spend around 200 le a week.
If you eat at Mcdonalds/KFC/pizza hut etc.. for 2 could be 50 LE(or more) for one meal.
I would budget around 1000-1500 le for food a month.

4) What things should we be sure to bring that we may not be able to buy there very easily?
Actually (this is my opnion) you can get most things in Cairo. If you have specific personal items that you use, creams etc.. tehn bring a stock, books that you enjoy(it can be difficukt to get good books here and can be expensive) Furniture/household good here, can be foundeasily and are cheap, depending where you shop.

5) Is it easy to receive shipping from the states?
I am not sure as i never shipped anything and tended to bring my stuff in luggage. However, i know that your better going to receive it yourself from teh airport, and clear customs. I believe they charge a fee, but dont know how much.
Comuter stuff etc... is easily accessible here.

Just to state, all the of the above is my own opinion, just in case i have things wrong.
One other point. Ramadan the holy month starts around August 23rd and lasts until end of Sept. During this time Egypt slows down and sometimes things take a lot longer to sort out. People are fasting during the daylight hours so some business's may operate on different times.
Enjoy your time here