Hi there!
Lots and lots of questions!
1) Let's discuss tickets first: you can book a cheap ticket (i.e. Ryanair or EasyJet) to Sao Miguel and then, as long as you don't stay in that island for over 24h, request the local airline company, SATA, for a free reroute to Faial or Pico (you can also reach those islands by boat, depending on how much time you have available).
About the free reroute check this link (do give them a call if you have questions): http://encaminhamentos.sata.pt/en/Inter-Island-connecting-flights-inbound
Reaching by boat from Sao Miguel (to be honest, I don't know how 'hassle free' this option is, but keep it in mind as an alternative): http://www.atlanticoline.pt/p/p/index.php
2) I wrote 9 posts (one for each island) outlining what to do in each of them, and recommended local companies (highest scores on TripAdvisor). Here are the ones about Faial and Pico to get you started on your research:
http://www.tripper.pt/travel/portugal/what-to-do-in-the-azores-part-7-faial-island
http://www.tripper.pt/travel/portugal/what-to-do-in-the-azores-part-6-pico-island
3) I think 8 days is more than enough and, yes, you'll need a car. You can either rent your own or book a taxi or a tour to show you around (the hotels usually have packages for this but if you're staying on an airbnb sort of solution ask around, there's always someone who knows someone who knows someone) ;)
In general, the Azoreans are welcoming and love to show off their islands. Most islanders live off the tourism industry, so making a good impression is part of our DNA.
Drop me a message or an email if you have further questions. I'll do my best to help you out.