Can the person who decrypts a homomorphically encrypted value know what operations were run?
No, not with SEAL at least. The noise is basically a measure of how much uncertainty there is after the operations are performed. Past a certain point there is simply too much and the decryption may be affected by it (the noise does not "round off" correctly), after which decryption may result in incorrect values. Because of this the "noise budget" is used in the library (invariant_noise_budget). Your results can safely be decrypted as long as the noise budget is positive.

It would probably be possible to gather some information about which types and how many operations (for some operations) occurred by analyzing the noise. Certain operations (rotation for example) do not consume the noise budget and would therefore be undetectable by a noise-analysis attack that depends on the amount of noise. Assuming SEAL does not have any huge flaws in it there shouldn't be any attacks that will let an attacker list out exactly which operations occurred.