There are 52 cards in a standard deck; 4 are Queens and 4 are Jacks, and Queens and Jacks are mutually exclusive, so you have a total of 8 cards that will satisfy the requirement. Therefore, the probability of drawing either a Queen or a Jack is 8/52, or 2/13.
For the game:
You have a 50% chance of rolling an even number on the die, and a 50% chance of getting a Head on the coin flip. The probability of both occurring is 50% * 50% = 25%. Therefore, there is a 75% chance you will lose the game. The expected value of this game is $10 * 25% + (-$3) * 75% = $0.25.
I'm not exactly sure what the definition of a fair game is, so I can't answer that question.
If you play the game 100 times, the expected value is $0.25*100 = $25. You would expect to win $25.
For the four children question:
There are 2^4 = 16 ways of having four children (since each could be either a boy or a girl), and there are 4 ways of having three boys and one girl (depending on which child is the girl). Assuming equal probability of having either a boy or a girl, you have a 4/16 = 1/4 chance of this event.