There is a coin worth $1 USD that doesn't require mining and can be exchanged anonymously. Is the cryptographic aspect a hard requirement?

http://en.wikipedia.org/wiki/Dollar_coin_(United_States)

I think it is a hard requirement; nobody keeps tight track of $1 coins because it's hard to trade a few million dollars using just that. A cryptographic currency wouldn't have this issue.

Is there a crypto coin where the mining work isn't a useless waste of energy?

Bitcoin mining (and maybe litecoin, dunno) doesn't do anything useful in the process of using all those hardware and electrical resources. No protein folding, no SETI, no looking for mersenne primes - just a bunch of crypto work that gets you bitcoins. It'd be nice if you actually had to produce some knowledge to get the coins.

This the common mistake. Mining does useful work: validating transactions.

How is finding the digits of 2^57,885,161 - 1 useful?

Many of the proponents of these currencies believe that a currency governed by mathematics is better than a currency governed by fiat. Since the government can print as many USD as it wants there's no way to fix an exchange range unless the cryptocurrency allows unlimited money supply increase, which clearly has to be controlled by some central authority.

To index to a fiat currency it requires a market maker. Some person or group that will guarantee that if you have $1 they'll give you a unit of currency and if you have a unit of currency they'll give you $1. To meet the demand for this currency it would require the market maker would need to be able to print money, otherwise the price could naturally increase without a way to stop it. Also, to be able to survive a run on the bank, for every unit of currency they issue and take in $1 they would have to keep that $1 in reserve in case everyone tries to cash out at once.

Letting a central authority print money that had no obligations to the general public is dangerous. Also the market maker would likely spend some of the money they take in. If there was ever a run on the bank they would just shut down the currency and keep the money that was paid in, which would make a run on the bank more likely.

This is very similar to online casinos that let you buy chips, but instead of offering gambling services, they would offer annonomous transaction services. Some casinos have just shut down and kept all the money or if it becomes a legal problem (as any solid fully annonomous currency would) could be shut down by their local government.

There is just no good reason to trust a currency that requires a central authority that can print money and that you don't trust and there is no reason to trust that they'll use the printing power responsibly.

If it is not mined, who gets the starting coins?

The inventor.