If they got home 10 minutes earlier, that means the wife saved 5 minutes of driving to the station and 5 minutes back from the station.  Therefore she picked up her husband 5 minutes before she would have usually meet him at 6PM.  He must have been walking 55 minutes.

Does he get paid the $8 regardless of whether or not he idles that day? That makes a difference in the answer

There are four possibilities for your neighbor’s children: BB, BG, GB, and GG. All of these are equally likely.

Once your wife tells you she knows one of the children is a girl, you have eliminated the possibility that the children are BB.  There are three possibilities left, and the remaining possibilities are still all still equally likely, so there is a 1/3 probability that both children are children.

However, when your wife says “The way I found out that at least one of the children is a girl is that I saw one of the children playing outside, and it was a girl”, the probability changes.  Using Bayes’ rule,

P(GG | child wife saw was G)
= {P(GG)P(child wife saw was G|GG)} / {P(child wife saw was G)}
= {P(GG)P(child wife saw was G|GG)} / {(P(BB)P(child wife saw was G | BB)) + (P(BG)P(child wife saw was G | BG)) + (P(GB)P(child wife saw was G | GB)) + (P(GG)P(child wife saw was G | GG))}
= {(1/4)(1)} / {(0(1/4)) + ((1/2)(1/4)) + ((1/2)(1/4)) + (1(1/4))}
={1/4} / {1/2} = (1/4) * (2/1) = 2/4 = 1/2.

So after the second bit of information, the probability the second child is a girl is 1/2, which matches our intuition

The answer is 8.

“I sit between you and a star”: The number 8 sits between the letter U (you) and the asterisk (a star) on a standard keyboard.
“but far below me is an empty space”: far below the 8 key is the space bar
“I can be infinite or finite depending on how you look at me.”: If you look at an 8 sideways, it’s the symbol for infinity.  If you look at it normally, it’s an 8, which is finite.