Hello!
1. Yes. Even if you work there as a freelancer, you should have at least a "freelance contract". A piece of paper that links you to the training organisation with signatures from both parties. Otherwise, if something goes wrong, they might just say: "never seen this man before!".
2. Never came across that. Is this something specific to Ireland?
3. The flying club! Have you got an idea what you will earn as a flying instructor? And how much this kind of insurance is gonna cost you?
Happy instructing,
Max