I'll Show You Young Punks How It's Done: The Bat-Man modifies his attack value by +1 when he's attacking a character whose current click number is lower than The Bat-Man's current click number.

The House Always Wins: Immediately before The Bat-Man makes an attack roll, you may choose a number between 1 and 6. If the attack hits and the chosen number is on one die, modify his damage value by +1. If the attack hits and chosen number is on both dice, instead modify his damage value by +2, and after actions resolve, heal The Bat-Man of 2 damage.

In the trait, what is meant by a "current click number is lower than The Bat-Man's"?
Find The Bat-Man's current click number. Let's say it is 4, for the sake of the example. Now look at the current click number of the character The Bat-Man is going to attack. If that number is less than Bat-Man's number, he gets the bonus. So, in the case of our example, if the opposing character's click number is 1, 2, or 3, The Bat-Man will get the bonus.

So how does the trait work with older characters that do not have a click number?
It doesn't. Without a click number, you have no way of knowing exactly what click the character is on (remember, you can't spin through the dial to check and cheat sheets and other reference material are illegal). So it just can't be used if there is no click number printed on the target's dial.

For the special power, does the healing only occur if both dice have the chosen number?
Yes. The part of the special power describing a single die, and its results, is one specific sentence and the part of the special power describing two dice, and their results (including the healing), is another specific sentence.

With the special power, I am not 100% sure which numbers I can choose. It says to pick a number between 1 and 6... does that mean I can't pick 1 or 6, since those numbers aren't actually between?
You may choose 1,2,3,4,5,or 6. "Between" is to be read as being inclusive on this power.

Edited to update the entry for the attack power.