Both "she stood there crying" and "she couldn't but stand there crying" are correct. The second sentence means that the only thing she could do was to stand and cry; circumstances made anything else impossible. The first sentence says nothing about what else she could have been doing.
''I hear someone shout in the distance'' is correct. It means you hear someone make a single call. If the person has been shouting for a longer time, you'd say "I hear someone shouting". Since usually you don't know whether it's a single call or continuous shouting until the shouting has ended, the most common usage of "shout" is in the past tense: "I heard someone shout". For the present tense, "I hear someone shouting" seems more common and natural.