There are 2 ways to leave act 1: the stairs behind a locked door you get the key for in Samuel's lair, or creating some explosives (using the coal and a couple other ingredients), and using them at a particular location. A book you can find gives the recipe for gunpowder, and a conversation you overhear gives a hint on where to use it (see
here).
For the Tree of Insight, the cutscene conversation is only triggered if you talk to Stignix in the imp village after you rescue him from the spider queen.
Do you mean 'magic rope' as in a belt? I don't recall there being anything useful there other than being able to ask the tree a question (and even then the best thing you can ask for is gold).
Beyond Divinity Strategy Guide