-get a job (certain identification needed for the company to satisfy immigration law requirements)
Find someone who buys and fixes up houses for sale or rent. They can usually use a hand with roofing, drywalling, electrical, cleanup, and all sorts of other stuff. A lot of them pay cash and don't require ID.
-open a financial account (it has been so long since I did this- is identification required for this, or would it be under the new rules?
OK, you got me on this one. I have no idea how to monkey wrench this one. The only way around it I know of is to not deal with checks. Cash, barter, and precious metals only. I've known people who distrust banks so much that they wouldn't have a checking account and would carry the entirety of their paychecks with them. If you're going to do this, though, getting mugged has higher risks of losing your dough.
-get a passport
If you never leave the country, this shouldn't actually be an issue. There might be a workaround for this, at least for some people, though:
"
To obtain a passport for the first time, you need to go in person to one of 6,000 passport acceptance facilities located throughout the United States with two photographs of yourself, proof of U.S. citizenship, and a valid form of photo identification such as a driver’s license."
Note that it says "such as". Student IDs, FOID cards, CCW permits, might work. Also:
If none of these are available , you will need:
1. Some signature documents, not acceptable alone as ID
(ex: a combination of documents, such as your Social Security card, credit card, bank card, library card, etc.)
AND
2. A person who can vouch for you. He/she must
* Have known you for at least 2 years,
* Be a U.S. citizen or permanent resident,
* Have valid ID, and
* Fill out a Form DS-71 in the presence of a passport agent.
Workaround. Neat! By the way,
link.
-get cold medicine (pseudoephedrine being on the verge of going behind the counter with ID and registration required to get it)
Um...don't buy cold medicine?

OK, some people just won't be able to stand being without something for their cold. Tylenol and lots of Vitamin C will help. I usually use Nyquil if it's so bad that I have to take something (Dayquil if I don't have anything to do, it makes me sleepy). Also, natural remedies do work for some people, so you might look into those.
-get a cashier's check (for example, if I wanted to buy a house)
Dunno if they'll take a money order for buying a house, but I've never been asked for ID to get one. Of course, most money orders can't be had for more than $1000, so carrying 20 money orders in to make the down payment for your house might look kind of silly. See if they'll take a student ID instead of a driver's license. Maybe a library card or even a social insecurity card in combination with a credit card or something like that. There are some people out there who don't have driver's licenses, but I have a feeling they still manage to get cashier's checks. Ask one of them how they manage to get around that.
-get a mortgage (was ID really required when I did that, or did I just give that up too easily) or rent a place
Again, alternate forms of ID will usually be accepted if you don't have a driver's license. Remember, some people can't or don't drive. There's bound to be alternate forms of ID accepted somewhere along the line. Birth certificate + social insecurity card + a witness or something. If you can get a passport using the above exceptions, you could probably use it, but passports are likely to be chipped in the near future, too.
-make certain firearms purchases
Don't buy from licensed dealers and buy ammo at the gun shows or a shop where you're already a regular. You may not get the handiest, dandiest guns out there, and you will wind up buying guns that have had a previous owner or two, but that's not always a bad thing.
-get a pilot's license
Ultralights. As I understand, no license is needed for planes that fall under the ultralight category.