fully homomorphic encryption with information-theoretic security? An encryption algorithm with information-theoretic security is one which even with infinite amount of computation cannot be broken. That is, given only the ciphertext, no amount of computation can reduce your uncertainty about the plaintext.

Fully homomorphic encryption is, roughly, an encryption scheme where $\text{Encr}(T(x)) = T(\text{Encr}(x))$ for arbitrary transformations of the ciphertext and plaintext (this is not entirely accurate but that's not important for this question).

Is it theoretically possible to have an information-theoretically secure FHE scheme? That is, is it possible to encrypt a plaintext $x$, have an external party do computations on them in a fully homomorphic way, without them being able to gain ANY information about the plaintext even with infinite? computing power?
You said you don't need to conceal the set of transformations that have been applied.  In that case a simple scheme is

$$T(\text{Enc}(x)) = \langle T, \text{Enc}(x)\rangle,$$

where $\text{Enc}$ is information-theoretically secure encryption (e.g., the one-time pad).  In other words, you simply list in the cipher text which transformations should be applied to $x$ after it is decrypted.