Why does BIP32 use the derivation path m/0'/0'/k' by default? What is the significance of this particular path, and why not just use m/k' or m/k directly?
The tree structure is used for organizing the keys you generate. Its just a suggested way to do it. The first number is the account, in this case account 0 (the default account). This let's you support multiple separate accounts with keys of their own. Then you have the chain, either 0 or 1. 0 is the external keypair chain, and is used for generating new public addresses. 1 is the internal keychain and is used for things like change addresses - the addresses you don't give out to people (hence 'internal'). Finally the k is obviously used to generate the k-th keypair for account and keychain you've picked.