It is no coincidence that these openings have been the favorite weapons among top GMs (Fischer, Kasparov) for a while now. The reason isn't that they are complicated though, it is because they offer great counter-attacking chances for black. This is mainly because of the unbalanced pawn structure that is characteristic of these openings. Opposite-flank pawn storms are common---harldy a sign for peace. Compare positions arising from the Kings Indian or Sicilian with openings such as the Petroff (1e4 e5 2 Nf3 Nf6 3 Nxe5 d6 4 Nf3 Nxe4). Black achieves equality often in this opening, but because of the balanced nature of the pawn structure, it tends to be drawish. The search for winning chances for black has been a long one, and it is commonly agreed today that the only way to achieve this is through unbalanced positions that offer chances to both sides.

You are talking about the Algebraic system of notation, now the world standard. I would very much urge you to take the time to learn this system....it takes less time than you might think to make it second nature. You wouldn't expect to be much of a Ham radio operator without knowing Morse Code, would you? :-)

Your Dad is right, the king CAN capture an attacker to get out of check. Whoever gave you that misinformation may have been thinking about CASTLING. A king cant castle while in check

This is a matter of taste, of course, but I would recommend the Indian defences (King's Indian, Nimzo-Indian, Old Indian) for the reasons stated above in the first question. Also the Benoni family, including the Benko Gambit offer Black great chances.