Changing the name of the Island.
You can informally call your Island anything you want (for example, many building owners give their buildings names that are widely used even if they are not officially recognized), but legally, the name of the island must be approved by the appropriate governmental body (Hawaii and/or the U.S. government).
Build an airport.
Build some houses for some people I know.
Bring hospitals in my Island( for any case it happen)
All of these things require that various government regulations be complied with by you. This includes land use regulations (i.e. zoning laws), building codes, laws pertaining to regulation of airports (from the FAA), and regulations pertaining to hospitals (many states require a "certificate of necessity" to build a new hospital).
Also pertinent would be state and federal environmental regulations which impose limitations on your water and sewage systems, your solid waste disposal systems, what measures must be taken in the generation of electricity, how hydrocarbons must be stored, what must be done to protect endangered species and preserve habitats, how toxic waste and nuclear facilities are handled, and so on.
Land use regulations and building codes in many places are the function of a municipal government or a county government, and if you formed a new municipality on your island, you would receive more autonomy in these matters, in exchange for the municipality having to have a city council elected by all eligible voters who live on your island and the municipality having to obey the limitations that apply to governmental bodies generally.
Hawaii has one consolidated city and county government (the City of Honolulu) and three county governments outside that City. So, land use and building code issues would be governed by whichever county you were a part of, unless you formed a municipality for this island. This said, in Hawaii, many functions usually carried out by local government elsewhere are handled by state government in Hawaii.
To put security forces to guard my island.
You can have a private security force, subject to occupational licensing requirements of the State of Hawaii, but they are limited to the same restrictions on the use of force that apply to private individuals. They are the moral equivalent of mall cops or store security or bouncers at bars.
Only a government, such as a municipality or a county government or the state or federal government, can have law enforcement officers who have the full authority that law enforcement officers have which private security forces do not (e.g. the authority to arrest someone on an outstanding warrant, to conduct a court ordered eviction or property seizure, to obtain a search warrant, etc.).
To put a law that everyone shall clean the beach and no throwing
miseries in the water.
A property owner can establish their own rules and establish fines for breaking them in the same way that many home owner's associations do. These would be enforceable ultimately, only by recording a lien and imposing a fine which would have to be collected through a court authorized by the government.
What is in my mind?
I was thinking that an owner, has the right to do whatever he wants
with his own things. So, I was thinking that, If I were to become
officially an owner of an big Island, then I have the right to do
whatever I like with it, because now it belongs to me.
You are mostly wrong. There are many things, often called "sovereign rights", that a government can do, but a private property owner cannot.
One helpful way to think about it is that all property owners, even property owners who have 100% ownership of land with no liens or mortgages or encumbrances, are really just tenants on land that ultimately belongs to the government. The government can charge you rent for your property (i.e. property taxes) and can impose all manner of regulations regarding the use of your property, just as a landlord would, subject only to its own internal constitution regarding how the government is allowed to act.
This heuristic understanding of the relationship between a property owner and the government is even reflected in the name of some kinds of property ownership. Property co-owners are called "tenants in common", "joint tenants", or "tenants by entireties" (and a sole owner of property was historically (and in rare throwback instances even today in some states) called a "tenant in severalty", although "owner of property in fee simple absolute" is more common terminology these days), even though they aren't formally renting the property from anyone, because they were tenants of the sovereign under feudal English common law legal theory.
If you are not an independent country, you cannot enact laws, legally ignore laws, punish people with incarceration or corporal punishment (not allowed under governmental criminal laws) for violations of actions, or use force to compel people to act (outside narrow exceptions found in criminal law).
It is quite easy to form a municipality if you have enough people on your island, but while municipalities can exercise many kinds of governmental authority that is not available to private property owners, municipalities have to follow the rules that apply to local governments in that state and country, the most important of which is that everyone who lives in the municipality, and not just the property owners, get to vote for the municipality's elected officials.
As sole real property owner in a municipality you still have a lot of power and can refuse to rent your property to people who you think would not support you in municipal elections. But, the municipality is still not you and you can't control it like a monarch.
Informally, complete control of a municipal government makes it very likely that the government will always exercise its discretion in your favor, which could provide near immunity from many laws, except that civil lawsuits in state courts which the municipality does not control, and county and state level law enforcement officials whom the municipal government does not control, also have the ability to hold the people who control the municipality accountable.
The Example of Lakeside, Colorado
One real world example which is quite close to the hypothetical one that you propose is the municipality of Lakeside, Colorado (land area roughly 160 acres, population 20 in 9 households), which has an amusement park, a college, some office buildings, a strip mall with some big box stores, and a couple of small apartment buildings just across the western boundary of the City and County of Denver near its northern boundary.
All of the land in Lakeside, Colorado is owned by a single women, Rhoda Krasner, after whom the Lake that makes up 20% of the town's territory is named (who inherited it from her father). Rhoda Krasner is about as close as you can get in the United States to being a hereditary aristocrat in a constitutional democratic arrangement, but as the explanation above demonstrates, even her powers aren't quite as epic as one might think.
Lakeside has a municipal police force which mostly enforces the law in the amusement park, and controls all of its own land use decisions. The town council is elected by the subset of the 17 adults (in 9 households) who are tenants in the apartments owned by the property owner who choose to vote in municipal elections. In practice, those tenants are docile and vote for candidates who support the town's sole property owner, but, in theory, they could elect a town council that would impose tough regulations and heavy taxation on the sole property owner.
The town incorporated in 1907, shortly before the amusement park opened and was originally called "White City" (incidentally, while municipal governments can't unilaterally name the geographic features in their territory, they can unilaterally choose their name and change their name, and are given great deference by other levels of government in choosing names for features within their boundaries), and ownership of all the land was purchased from a member of the Coors family by the father of the person who currently owns all of the property in the town in the 1930s.
There are several other examples of similar arrangements historically. A number of municipalities in the United States were "company towns" in which all property in the town was owned by the leading business in the town (often a mine or factory), the only stores in the town were owned and operated by that company, and everyone else in the town was a tenant of the company and had to abide by its rules.
Sometimes, to avoid the possibility of a "coup" by residents, that is inherent in an operation like Lakeside, Colorado, land is deliberately left unincorporated and served by "special districts" with much narrower regulatory authority in which property owners are often allowed to vote as well as residents, and home owner's associations or the equivalent. A county government may have authority over the area, but in some states, county governments don't have a lot of regulatory authority and may need a warrant to enter into privately owned property without the owner's permission.
Many government owned universities are likewise very dominant in managing the affairs of people affiliated with them and can even have their own police forces without being government by a body elected by the residents of the university, since they are a division of the state government whose officers are appointed by officials elected at the state level.
At least one medium sized city in India (with all land owned by a corporation) and at least one small city in Africa (owned by a church) are operated on a similar basis. In the city-state of Singapore, until very recently, almost all land was government owned, and most so called land owners there, even today, have property rights that fall far short of those of land owners in most of the rest of the world.