When a dice or all-in poker bet is settled but there is still the "open for additional betting" info below, does that mean you can make another bet with the same parameters and the result will be randomly determined again?
Yes. You can send multiple bets to the same bet address. Each dice roll will be different and random because it uses the transaction ID of the bet transfer as the seed.
By the way, you can also roll the dice multiple times in a single bet transfer when sending from wallets such as blockchain.info, that allow you create a transfer with multiple outputs to the same address. The dice will be rolled for each output separately.
This is also supported for Sport bets. For example, you can order the bet in advance once, then send multiple bets to the same bet address while watching the game. Each bet will confirm at the current odds at the time of arrival, which will fairly reflect the situation in the game, and you can optionally specify the minimum odds that you are willing to accept.