Your assessment of (1) is correct. There are 52! permutations without replacement that can occur with a deck of 52 cards. It should be clear that the first card can come in 52 possible ways, the second from (52-1), the third from (52-2), ..., etc. Thus, we have 52*51*...*1 = 51! permutations of the entire deck. The case of a completely ordered deck is just one of those 52! and therefore the answer is, as you indicated, 1 / 52!
As for part 2, you need to be clear about what the probability is about. You do not want to commit the Gambler's Fallacy. The probability found in (1) will not change just because we shuffle the deck, no matter how many times we do it. The odds of a given hand in poker, for instance, does not change if the deck is randomized each time (and that is the key here). However, if you're taking the experiment to be "what are the odds of getting a completely ordered deck in n shuffles" then we're looking at a binomial experiment. In each of the n trials you either get it or you don't. The probability of a success comes from (1). From there it is a direct application of the binomial distribution.
Therefore, your friend is correct in that "the probabilities do not change for any specific shuffle." That is what it means to have determined (1). But that is not the same thing as talking about observing a certain number of successes (in this case, one) in n such shuffles. Each shuffle is considered a Bernoulli trial, and as I indicated above, the binomial distribution is the extension of those trials. The probability of a joint event as represented in a binomial distribution is clearly going to be different than any one Bernoulli trial.