1. Is it ok to use quotations by famous people in my task 2 essay?
I wouldn't recommend using quotations. Unless you have thousands of them in your head, it's unlikely that you'll remember a relevant one. The examiner is likely to judge a quotation as 'memorised language'.
2. How do examiners know whether I'm using 'memorised language'?
All of the language in your brain is memorised, so I should explain this properly. The examiner would be impressed by a memorised phrase like "greenhouse gas emissions contribute to global warming", if the question asked you to explain an environmental problem. However, examiners are not impressed by the kind of phrase that students think they can use for any question e.g. "This is a controversial issue that has led to heated debate among people on both sides". Can you see the difference?
3. Can you help me with apostrophes (e.g. student's or students')?
You really need a grammar website for help with that. Try this one.
4. Should we use different language in the speaking and writing tests?
You can use some less formal language in the speaking test. Here's an example.