how to change node edit path from node/[nid]/edit to node/[nid]/update?
I think this is a legit request as long as you add an additional alias as opposed to overwriting system paths.

I would implement hook_node_insert() and then use path_save() to set your custom edit path on node creation. You will also probably need to make sure that this hook runs after the Pathauto module has set its path, so you can use the alias it sets as part of your additional edit alias.