Why did Pavel Young accept Harrington's challenge?
Pavel the political figure had no choice. Had he not accepted, he would no longer have been a political figure... period. So, yes, he could have refused, but only at the price of giving up everything he felt he had left in life. If you'll notice, he didn't exactly cover himself in steely-nerved glory when the moment came, and that was largely because his political ambition (i.e., hunger for power) had gotten him into a situation he lacked the intestinal fortitude to face up to. And remember also that he had agreed to a protocol in which he only had to face a single shot from Honor. Yes, as the moment loomed large before him he became more and more aware of his own mortality; at the moment he actually accepted her challenge, the instinctive need to preserve his position of power (and to avoid a situation in which no one in "society" would ever so much as acknowledge his existence once again) overpowered his fear that she would be able to kill him with that one shot. In the event, his nerve snapped, leading to his ignominious demise.
As for why none of Paul's family members or HH's friends challenged Denver, there were two reasons. (1) No one knew where to find him until Georgia slipped the word to Ramirez and McKeon through an intermediary, so no one could challenge him, and (2) Would you really want to be the person who challenged and killed him instead of leaving him for Honor to deal with?