In university, we never discounted puts and calls strikes in order to determine payoffs. My question is this: sometimes the question provides you with the risk free rate, do you need to discount the strikes by it (according to the number of motnhs)?

depends on the question: -if the question is related to the putcall parity then you will need it. -if the question is asking you about euro options - you will ALWAYS need to do it… (due to the nature of euro options) -if the question is about american options - you will need to apply it as it comes by, eg if they ask you what the min value of an american put is and they give you a disc rate then dont worry about it.

Yes

You should not discount strike to determine the payoff because the payoff is paid at expiration. However when determining the current value of the option (e.g. max or min value) you should discount.

There is no scenario in which you should alter the strike for a vanilla call or put. The strike is part of the contract’s definition and is constant. If you are calculating the payoff at maturity, discounting is not needed. Before expiration, you don’t know the payoff yet. So, to find the contract’s value, you use the Black-Scholes formula, which needs various inputs, including a fixed strike.