When all the cards are out and betting is done, which player is required to show their cards first? Does it vary based on whether or not there was betting in the final round? Is there a standard accepted rule or is this one that varies from casino to casino?
This rule will vary from house to house. The two options are:
In any case, after the first person shows, the onus to show (or muck) proceeds clockwise around the table.
Which is equivalent to option 2 above.
In a number of casinos, I have seen the rule enforced that players only need to show their cards if they are seeking a claim on the pot. This practice is why many live players will verbally declare "you have me beat" without exposing their cards, in hopes that their opponent will show and take the pot without further incident. The player making that declaration may even go so far as to discard their hand into the muck.
The only time this practice creates a major problem is when someone at the table suspects cheating of some sort. This is where the live rule emerged that made it acceptable (if frowned upon, since you are essentially accusing the other players of cheating) to request to see the hole cards of all players who were in the hand at show down. That rule, in turn, is what led to the common practice of online poker sites showing all players hole cards in the text of their hand histories.