1) Are you SURE that's the Generator Matrix? Perhaps it's the Traspose of the generator matrix.
2) The Parity check Matrix is created by simply transposing the Generator Matrix and "OR"ing consecutive pairs of rows. For this reason, the Parity Check Matrix should have one row less than the Generator Matrix has columns.
If I actually knew anything about this sort of thing, I might be able to answer "C". You share your thoughts and maybe I'll do some more reading.