You've got quite a high-rise project here, which definitely calls for a contractor who knows what he/she is doing and is equipped to handle the job.
As to materials selection, I may not be the right guy to answer this where cost is a MAJOR factor, because I will stand firmly by the idea of "Do it right, and you'll only do it once." My strong recommendation for low maintenance & longevity will be to use a Trex or similar product for your decking surface. Not one of the many decks I've built using Trex has ever been reason for a client to contact me with a problem.
Initially, yes, the cost will be more; currently just below the cost of using redwood in my region (midwest), but when you are looking for low, or no maintenance, this type of product is a true performer.
As to your framing, this might depend on what area of the country you are in & the size and spans of the deck/gazebo you are creating. Most often I will use a treated lumber for general framing & joists, but not for any finished areas or where it has direct sun exposure.
It sounds like you may be needing some quite tall support members, too, for which I'd not use treated lumber if the supports will need to reach 15' plus railing height uninterrupted or otherwise locked in with other framing; you may wish to look into steel posts for your supporting columns in this case.
When it comes to choosing who you will trust to complete this project, definitely you will want to have references, call those references, and if possible arrange to go see some of the work that has been done for others. Don't do this through the contractor, speak directly with the references. Preferrably, this contractor should readily provide you with 6 to 10 references with somewhat similar projects, and be quick to provide more should you ask. When I am asked for references, I supply my entire customer list of names with references as to the type of work that was done for them, and say "Take your pick."
You can also check with the Better Business Bureau for any complaints on record, but don't rely just on a clean bill from the BBB to mean the person is honest & will do a good job; in this respect the BBB usually has no clue.
Ask if they are licensed & insured, and ask for a copy of their insurance certificate. Be sure they plan to obtain a building permit. Be sure the terms of payment & job completion are clear, and read all contracts very carefully before signing.
Good judgement in "reading" people will keep you out of most trouble, but always go the extra mile to double-check someone if you're not sure. If your contractor quickly proves to keep his word, you can usually avoid the necessity of an adversary relationship during construction, such as having to follow up with assurance of obtaining permits, meeting codes, and providing the quality & services promised. When in doubt, ask; a contractor who's doing as he/she should will gladly explain any process you might have question about.