Regarding Turing completeness, I have the math background to answer:

Saying that a programming language (whether AT, or Ethereum, or C++, or anything else) is "Turing complete" means that it can perform all the calculations that a Turing machine can perform. A Turing machine is a technical math/CS concept, but the key thing to take away for this discussion is that Turing completeness only says something about the calculations that can be performed. It doesn't say anything about actions.

Turing completeness is a good thing! It means ATs can read data sent to it and add, subtract, multiply, divide, compare this data, and perform all other computations that you'd normally expect a computer to be able to do.

Bad AT behaviour may indeed be possible, from actions like e.g. spamming AMs. But Turing completeness doesn't say anything about actions like sending nxt, or composing and sending an AM. These would be additional features of ATs. It's up to Ian and his team to decide what actions ATs should be allowed to/not allowed to do, to guard against bad behaviour.

As I understand, Ian's writing a virtual machine (VM) for ATs. That means a place set aside for ATs to perform their calculations, and that means ATs won't be allowed to interfere with the rest of your computer. I think ATs will also only be allowed to send transactions from the account that they're "assigned" to, so that's another good restriction.