Questions & Discussions (about the course or NLP in general)

Your $P(w|C)$ are wrong: this is not the the proba that a document (of that class $C$) contains the word (nor contains that many words (count)), but the probability that this word appears in that class, which can be estimated (MLE) by number of occurrences of $w$ (in $C$) divided by the total number of words in class $C$.
For instance $P(\text{cat} | \text{Space})$ is estimated by $\frac1{10}$, $P(\text{Dog} | \text{Space})$ by $\frac4{10}$.