I’ll answer your earlier question here as well.
Regarding materials, much of the cape is very remote, and access to those materials is generally through having them transported to the area. I would suggest you investigate materials that would be available in the larger towns and cities near the Cape.
And your other questions:
1. Is there any company or organizations at Cape York that have a trained team and be responsible for building constructions or facilities at Cape York?
A: You can assume that if the appropriately skilled tradespeople are not employed by the relevant local stakeholders, that they will be able to hire workers from nearby larger towns and cities. A consideration for your design could be its complexity; could it be assembled and maintained with little technical knowledge?
2. How long from a design is submitted to the local Cape York authority to receive the construction permission?
A: This would likely depend on the design, and the zoning. I would suggest searching those relevant local and state government agencies to understand the administrative and regulatory requirements for your design. A starting point would be here http://www.cook.qld.gov.au/ and here http://www.cylc.org.au/about/
3. After building a construction, do we need to invite local authorities to test the facility?
A: It would be good to assume that an inspection to ensure safety of the design and adherence to any regulatory requirements will need to be undertaken.