[1] "Bikkuri" = OMG!
[2] "inai inai ba" = the peek-a-boo game you play with babies.
[3] "bigi ni youette" = a pun off of Atobe's catch line "ore no bigi ni yoi na".
[4] "dono" = lord, even higher address than "sama".
[5] "Sunmason" = Ibu's unique way of saying "sumimasen".
[6] His name is Kyuuki Kiichi, and no, you don't remember him.
[7] The attempted phrase is the phrase "kakinoki chuu no kyuuki desu", "in the persimmon tree are nine spirits"... Kyuuki in his name means literally nine spirits.
[8] "mou ii kai?/mada da yo!" (Are you ready?/Not yet.) is how the Japanese version of hide and seek is played. Ryoma doubles the "mada".