There is a testnet that works right now, but should not be used in production yet (I think it gets reset regularly? Not sure.). Within a few months, it is expected that the first version of "the real thing" will be up, but I don't think putting things on it quite then would necessarily be a good idea (probably depends on what things one is considering putting on it), because it is set up so that after a certain time, there will be a change to a new version, and while all balances will be carried over across that change to the new version, other parts of the state of the system won't be, so anything built on it would have to be set up again. (which could just be a matter of sending the same message again, so it might not be that much of a problem to set it up before the change I guess.)

Also, its probably important to note that a contract is run by every miner (or, every miner on a certain part of the network, depending on how the scalability thing ends up being done?), and that as such, you will want to keep any computation done by a contract to be computationally cheap, so that it will be cheap to use. Instead of having the computation do the computation that needs to be done, it may in some cases work better to just have the contract verify the accuracy of computations that have been done.

However, now that I think of it, I'm not sure that Ethereum would solve all of the problems OP gives, because one of the problems OP wanted to solve was that they wanted to demonstrate that they were not storing information that they shouldn't be. But with Ethereum, because the contracts are executed by "everyone", everyone has to have access to the data the contracts are using to run, and there is no way to insure that they don't hold onto that data.

One way to solve this could maybe be doing computations on shared secrets (as talked about in one of the Ethereum blog posts, but which is not something Ethereum is to have), but this might require more messages to be sent over the network than one is willing to use. Still much more practical than homeomorphic encryption though I think. (If one was using the shared secret thing, I'm not sure one would do it with Ethereum.)

Ethereum could/would solve the problem of ensuring that the program being run is exactly as claimed, but it might not keep certain information private. Depending on the specific problem at hand, there might be ways around that though?

EDIT: ok, so, something suggested that homeomorphic encryption might have gotten an improvement to the point of practicality recently? I wasn't aware of that. May make this post slightly out of date.