Have some questions, please help

1.) When calculating the value of an option using a binomial model (say a european call option on a bond), we essentially discount the futue payoffs/intrinsic values to find the option price. However, when valuing an otherwise identical american option the text says to use the higher of either the option value of the intrinsic value.

My question: how can the intrinsic value be higher than the option value? I didn’t think this could be the case bc the option includes time value which intrinsic value does not. Unless this is merely highlighting that the option value for an american option can be higher bc it is exercisable (which of course makes american options more valuable than euros). So in this case, the intrinsic value is the actual value of the american option which is not equal to the Euro option.

2.) How is time value captured using the binomial model if we’re discounting the intrinsic value at expiration ?? Is it bc the prices at that point reflect the future volatility and thus the time value is inherent in the intrisic value/payoff at expiration?

3.) Can someone help me better understand this concept, from the text: “in the binomial model, the hedge portfolio is riskless over the next period, and the no-arbitrage option price is the one that guarantees that the hedge portfolio will yield the Rf rate”.

Thanks, all !!!