Ah.. the CM/Classical question. This is one with a multi-faceted answer!
At HOD we definitely lean more toward the CM-style than the classical. As the ladies have already pointed out CM and Classical have some definite areas of agreement and overlap (most pointedly the reading of more "classic" type books), however the main thrust of the two philosophies is different.
CM-style readings focus on fewer books read more slowly over time, in essence fewer/more quality books done more deeply. Classical enjoys using fact books (i.e Usborne, DK, and other encyclopedia like books) as reading material and also encourages reading of abridgements of classics early on. CM readings are always living, using fact books only for reference, and recommend waiting to read the classics until the unabridged versions can be attempted. Not all classics are considered "good reading" in a CM style education. Classical often focuses on the "Great Books".
CM focuses on narration as the primary method of comprehension. Classical also uses narration, but more for the purpose of learning to summarize. In CM-style narration, kiddos are to borrow words from the author to retell the story. Narrations are often lengthy and detailed and there is no "one right" answer or certain key points that "should be" in the narration. This is the way the child connects to and makes sense of the reading. Classical narration is looking for a more succinct narration with "certain" main key points. There is more of a feeling that a good narration should have these "key points" in it. This is a different form of narration from CM-style narration.
CM and Classical both use dictation, but with different methods. CM uses studied dictation (meaning kiddos study the passage first to fix in in the mind prior to having it dictated). Classical also uses dictation, but does not have kiddos study it first. It is more of a test of what the kiddos know, rather than the practice of fixing it in one's mind.
Both CM and Classical use copywork as a form of early writing practice. This is an area of agreement.
CM uses delayed formal grammar instruction and delayed formal writing instruction. Classical education focuses on early rigorous grammar instruction and also on a rigorous schedule for memorization. CM also does memorization, especially of Scripture passages and poetry, but is not nearly as rigorous as Classical.
Both CM and Classical use written narration, yet the expected outcome is different. Classical uses written narration as a tool for learning how to write a summary through narration. There is more of feeling that there are certain key points that should be included for it to be done correctly. CM uses written narration as a tool to learn writing style by borrowing the author's style and wording to convey the thrust of the reading. It is not meant as merely a stepping stone to summarizing, as classical uses written narration. In CM-style narration, the student is instead trying on various styles of writing using the author's style, until they eventually begin developing their own style of writing.
CM and typical Classical vary quite a bit on their approach to Bible study and integrating God's word throughout the school day. CM believed this was foundational to all learning. Classical devotes very little time during the day to this topic, unless you follow a modified classical approach (such as the Bluedorn's Christian Classical or Memoria's Press's Christian Classical).
Character training and formation of habits were a huge part of CM's focus. She devotes much of her 6 volume series to those topics. It is in these areas particulary that I agree with her. The formation of a child's character and his/her habits is an overlooked topic in Classical education, as the pursuit of wisdom, knowledge, and education is supreme.
CM felt poetry study, nature study, art, and music were important. She studied science through nature, art through picture study of famous paintings, music through listening to classical pieces, and poetry through daily reading of classic poems. Classical looks at these areas as an "add-on", until they are done in the upper levels along with the rigorous study of history. With a truly classical schedule there is little time left in a very rigorous school day to devote to these things.
You can see that at HOD we fall on the CM side for almost all of the things I've listed in the above post. The two areas we have not included yet, are picture study and classical music. However, we have discussed these areas on the board quite a bit, and do add them in with our own kiddos. In future guides, we will hit each of those areas at least once (preferring to do famous artists, composers, and hymn writers when they best coordinate with our history study).
Anyway, from the description above you can see some distinct difference between the two approaches. When reading it, you may hopefully be able to sort out the differences and where you fall philosopy-wise a little better.
As far as "It Couldn't Happen", we have decided against using it with our own kiddos due to the "old earth" perspective in the book. Answers in Genesis had some notes about that particular book that we agreed with when perusing it ourselves.
With the deeper "logic" stage Bible study part of your question..., we have sought deep discussions and thinking about the Bible from the beginning with our guides, so we will continue in that pattern of weighing everything against God's word with our future guides.