1) Definitely British Council, everyone will tell you they are the same but, me and my friend took ielts from british council and IDP respectively, we had same english skills I got overall 8 and she 6.5. Then we discussed and she found out that my paper was far more easier then hers and even the marking was lenient. she also pointed out that British council paper was far easier then the sample papers. She gave again her paper from British council and got 8 bands.
2) Its both the same no difference whatsoever.
3)I do not know about that if this rule applies to all canadian test centres, ut all I know is that in India British Council gives you personal wireless headphones and IDP gives you headphones with wires. I have no experience with a speaker so I'm afraid can't help you there.
4) You will definitely get a preparation book once you register for the test from IDP/British council, it provides you with the best guide on IELTS. For additional help you can buy Cambridge IELTS by official cambridge press, its more detailed and vast. Apart from that try to speak as much english you can, read as much as you can, newspapers and magazines , if possible a novel. This will help you improve your vocabulary and let you get accustomed to how to use certain words in what context and what grammatical form. There are some online sites also which have very helpful tips.
if you are a decent speaker of the language then 1-2 month's will be sufficient to get a good score.
The most challenging are writing and speaking and key to good score there is practice and practice. Speak and write as much. converse with someone who will also correct you if you are wrong somewhere. time management is most needed in writing so take care of that.
hope this helped.