You should have received my email already with the documents. I will answer some of your more specific questions in the order you are asking...
Azores "soaks" in its European identity, with a noticeable independent twist, based on being 900 miles from the mainland of Portugal. The culture you are interested in is authentic and has been passed down from the 1400s. The traditions, though they vary a little between the islands, are pure and simple. The best times to experience the culture and traditions are between early June and late August. There are countless festas (feasts) among the 9 islands during the summer, and most have a religious inspiration. Azoreans pride themselves on their roots as islanders, and consider themselves "Portuguese" second. There is no tourist "show" that they put on, as they are normal people living their lives, as you would in your neighborhood. But they are extremely welcoming and honored to let you in and experience their culture. By now you have figured out by my words and your research that visiting the Azores will be like visiting no other European destination, and it will be very different than visiting mainland Portugal. The "draw" I believe is definitely nature, relaxation, history, gastronomy, the ocean (above and below), AND the people. Some things you will see and ways of life have not changed in over 500 years.
I have visited many islands around the world, including Hawaii, and would say just about every single one is there to offer a service and gladly take your dollars. Not so in the Azores! Honestly. You are considered almost family when you are there. They go out of their way to make you feel at home. You will be sincerely welcomed wherever you go. If you speak English, they will respond in English. If you try a few words in Portuguese, expect a big smile and a congratulations for speaking their language.
As a visitor, the costs are not expensive. Car rentals are comparable to most places. Hotels are cheaper, dining out is much cheaper than most cities, entrance fees are way cheaper, supermarket costs are comparable or lower. Tour companies, although I have never used theses services, are lower than most tourist destinations. Taxis are plentiful and drivers are known to be honest. If you want to buy a house, then it is comparable to prices in the United States.
So how can I give such a raving review? Well, I married into an Azorean/American family over 25 years ago. I have visited the Azores over a dozen times to visit family and plan to do so every year. I have traced my wife's ancestry back to the 1500s on the island of Sao Miguel and the 1600s (so far) for Santa Maria. We know her ancestry for the mainland of Portugal way back to the 1200s-1300s. I live in a city that is mainly populated by Azoreans who immigrated to the USA in the 1960s and 70s. I learned to speak and read the Portuguese language.
I hope this helps and you plan a visit. I could go on and on with answers and recommendations.
Take care,
Carl