I asked about the randomness of its appearance as the last digit of a block hash.
As long as the miner (or pool) doesn't have a significant incentive to ignore a successful block hash to continue trying additional nonce values for a different hash, it is random.
Whether increasing difficulty has any effect on it etc.
Difficulty has no effect on it.
Your chance to guess the right digit is 1 in 16.
That's not always true for this specific game as the a player can chose multiple options at a time.
You didn't explain what the options were.
You asked "how random is the last digit of a block hash really is". The chance of guessing a single random last digit is 1 in 16 if miners (or pools) don't have a significant incentive to ignore a winning hash to continue trying additional nonce values for a different hash.
Mining pools probably can not force the last digit of a block hash to be zero or any digit of their choice. It should depend on the nonce. Otherwise SHA would have been broken.
Sure, but if the block reward is only 12.5 BTC (like it will be in about 7 weeks), and the prize for guessing the correct last hash digit is 200 BTC, then it might be worth it to the miner (or pool) to just ignore a successful hash and continue to try additional nonce values to see if the next successful block hash has a winning digit. They might miss out on the 12.5 BTC since they didn't broadcast their block, but 1 out of every 16 times they do it, they have a 6.25% chance of their next successful hash earning them 200 BTC.