repeatedly hashing result and using reverse sequence as RNG secure?
A: No, an attacker can not guess an earlier number. That would amount to a preimage attack for the hash. S/he can not even gain a sizable advantage towards such guess. Arguments (not proofs)


Assuming $\text{secret}$ is full-entropy, the first hashing causes a sizable entropy loss ($0.827\ldots$ bit, see this), and each further hash causes a further entropy loss, but by an heuristic argument, after $k$ iterations, only $\approx\log k$ entropy is lost, thus plenty enough remains (for SHA3-224 and a million millions iterations, >195 bits).
Because it is added a counter rather than a constant, the $N_i$ falling into a short cycle becomes next to impossible, rather than very improbable.




B: The first number disclosed $N_k$ should come with it's index $k$, so that verification of the next number disclosed $N_{k-1}$ is possible by checking $N_k=\text{SHA3}(N_{k-1}+k-1)$ without guesswork on $k$. You can't manipulate the sequence once $N_k$ and $k$ have been shared with others. If you could, that would break the collision resistance of the hash.

Without $k$ published you would have a narrow possibility of fraud: on the second day you can publish $N_{k-1}+j$ and pretend $k$ was $j$ less than it really is; you wont be caught until the next day, and can't be caught if $j=k-1$.

Note: you could manipulate the $N_i$ when choosing the secret, e.g. so that $N_{987}$ is odd; and of course you know all the $N_i$.