Can the "Auto Select" hero complete dailies for specific hero types, if matchmaking happens to choose a hero applicable to a daily I have? Or do I need to specifically choose the appropriate hero ahead of time?
Yes, there's no problem in using auto-select to complete daily quests. If, let's say, you have a quest that wants you to win three games with a Diablo hero, you get a Diablo hero three times in a row through auto-select and you win the games you will still complete the daily. Daily quests are based on the games you play, it doesn't matter how you chose the character.