Not sure why the lower bounds for European put valuation could be P-(x/(1+rf)^t).
More specifically, I am not sure why we have to discount the strike price. What does that represent?
It represents the fact that you cannot exercise the option until expiration; you don’t pay the strike price until then, so today you have to deduct only the present value of the strike price.
with regards to the american put option, is it because the option can be exercised at any time prior to expiration , the strike price is not discounted?
Yes.
Other than what you stated above, does discounting the strike price to the present value (for European options) represent a direct relationship between time and the value of the option (that time makes the optino more valuable)?