Elliptic curve cryptography related key attacks
The answer depends on your definition of security:


Anonymity: this scheme will produce key pairs, and hence also public addresses, that are indistinguishable from random to anyone who does not know the hash function contents, if the hash function is modeled as a random oracle.
Cryptoeconomic security: People will not try and game your scheme if you can ensure that the value of gaming your scheme is less than the cost of doing so. In your example, if you use the blockhash within the protocol, an adversary controlling the blockhash may withhold the block they have successfully mined, in order to give themselves a better than average chance at winning your game. 
With this knowledge, you can find the lower bound on how motivated and powerful an adversary attacking your scheme must be by defining a security parameter $n$ such that:
$$ \mbox{value gained by winning your game} &lt; n * \mbox{bitcoin block reward}.$$


For your question of whether creating signatures with 'related keys' is secure, this is something I'm currently working on formally proving. With the hash function modeled as a random oracle, I believe we have the result that one can produce: $$a' = a + H(S),$$ and hence spend funds associated with an account if and only if they know the $A$ such that:
$$ A' = A + H(S) \cdot G,$$
with $G$ the generator of the elliptic curve group in question. However, this is dependent on your opinion of hash functions as random oracles. $H(i) = i$ would definitely not be sufficient, and $S$ is assumed to be secret.

EDIT: If $H(S)$ is not secret, there will be a de-anonymising attack on all transactions associated with your account, but people will not be able to forge transactions from you, or spend the bitcoin from that account, as they will not have the root private key $a$ necessary to form the private key. They will simply be able to scan the blockchain and group the accounts associated with you. This is how view keys and spending keys in anonymous cryptocurrencies such as Monero work.