I completely understand why you're confused. Opinion on this can range from the Queen has near absolute power to the Queen is a pointless figure head. I think how people feel about the monarchy colours their opinion on this issue as well, therefore you will never get a truly unbiased answer. There is so much about the monarchy that people believe to be true because of tradition or popular myth as well, for instance people believe that the Queen must be politically neutral, that she can not vote-yet there isn't a specific law that says anything like that, however even the official website of the British Monarchy mentions the Queen's resolve to be politically neutral.
The truth is that the Queen does indeed have massive powers, these are powers that are based in specific laws, powers that have not yet been restricted by any written law. She can dissolve parliament, she can sack a prime minister, she can break a treaty, she can declare war, etc. The only specific laws I know of that restrict her power are that she can not raise an army on her behalf, that she can not raise taxes on her behalf, and that she can not make and enforce any new law.
However, one should understand that despite the Queen having these powers she reigns by popular consent. She knows that monarchies can be overthrown if a monarch becomes tyrannical. Overtime her powers have, shall we say, gone dormant, they are never used anymore unless on the advice of her ministers. By convention over time it has become common agreement that she isn't supposed to use these powers unless for some reason it becomes absently necessary. The difference between someone who says the Queen has near absolute power and someone who says she is a pointless figurehead is based on whether they understand first, that the majority of her powers haven't actually been restricted by any specific law, and next, on whether someone believes she could actually still use them.
The rest of your questions are a bit easier to answer. The Queen does pay tax. She does have the right to rule simply because we haven't said otherwise (popular consent). She does not take any tax money for her own gain. The British people are technically subjects, but because that word is so old fashioned and because it has many negative connotations we have taken to using the phrase citizen instead.
A concept difficult for those who don't live in a monarchy to grasp is that she is the embodiment of the nation. For instance in America they consider their written constitution to be the embodiment of their nation and they salute a flag. In the UK she is the nation, she personifies it, almost as if she is a symbol rather than a human being. This means that laws are written and enforced in her name, and that people are tried in court in her name. Because of this it would be awkward to prosecute her for any crime, however understand that Charles I was also the embodiment of the nation and yet when the monarchy was briefly overthrown in the 17th century he was still arrested, tried, and executed, therefore while the Queen is technically above the law she is still bound to it. If she went on a killing spree she would either be forced into abdication or the monarchy would be overthrown, then she would be arrested.
As for land, for that you need to research the words: crown estate and civil list. Technically she is the largest land owner, however she allows the government to manage and profit from the lands she owns. George III agreed that the government could manage and make a profit from lands he owned (the crown estate) and that in return the government would give him a stipend (the civil list) this arrangement still goes on today, however each consecutive monarch has to agree to the idea and it isn't known for sure who would receive the crown estate in the event of the monarchy dissolving. The tax money the Queen receives (the civil list) is less than what the government profits from due to the crown estates, therefore one could say the monarchy costs nothing. As for as the civil list is concerned, this isn't the Queens pay slip for her to enjoy as she sees fit, it is money she uses to fund specific head of state expenditures.
You might want to visit the official website of the British Monarchy, and google civil list, crown estate, and head of state. I'll copy and paste my usual answer to similar questions like this below.
The Prime Minister is the head of government (he runs the government), while the King/Queen is the head of state. The Queen's function is to act as head of state.
What is a Head of State:
Heads of state and Heads of Government have two totally different roles in government, however in America the head of state and head of government is the same person (Obama) but in Europe it is preferred that they be two different people. In European nations without a monarchy a politician is usually chosen to be head of state and they're called a President, while a Prime Minister remains the head of government. Therefore, if the Queen didn't exist someone else would have to do her exact same job. If the monarchy wasn't here someone else would have to do the Queen's exact same job, this person would need the exact same amount of money.
What does a Head of State do:
The head of government runs the actual day-to-day agenda of the government, while the head of state is the ceremonial leader of the nation, the armed forces recognize them as commander & chief, they lead the nation in important events or ceremonies, they lead the nation in charity and good deeds, they pass out honours or awards, and most importantly they represent their nation abroad.
Comparing this to your nation (I'm assuming you're American):
When Obama tried to win Chicago as the Olympic host city that was a head of state role, when he welcomed and entertained the President of India recently at a state banquet that was a head of state role, when he and his family give support to charity initiatives that is a head of state duty, when he 'pardons a Turkey' each Thanksgiving that is a head of state role etc.
Why it works:
There are 10 nations in Europe that use monarchs as their head of state; it isn't just Britain, there are ten in Europe alone that do and these are arguably the best nations in the world to live with the highest democratic values, the freest democracies, and the most stable governments; places like Denmark, Norway, and Spain. It actually works very well. It works well first, because they aren't a politician (people hate politicians) it works well because they have their whole life to learn and prepare for the role, it works well because it is cheaper, it works well because unlike a politician they can not be bought or pressured into anything, and most importantly it works well because the most powerful politician in the nation (the Prime Minster) is forced to stay humbled and can't become a tyrant.
As for the money given to the Queen, it is called civil list money, but it is NOT a paycheque, nor is it her money to do with as she pleases. This is money that she uses to fund specific head of state expenditures. When the President of South Africa was recently in the UK for a head of sate visit it was civil list money that paid for his meals, travel, security etc. It is civil list money that pays for the Queen's travel when the government sends her on an official head of state visit. It is civil list money that pays for palace repairs and it is civil list money that pays staff. If the monarchy did not exist a politician would have to do these things and he or she would need civil list money for the same exact costs. If the monarchy was abolished the money given to the Queen now would not then be used instead for cancer research or benefits for the poor; it would just be given to a different head of state. Furthermore, the Queen is given far less to fund head of state expenditures than the presidents of Germany, Russia, Italy, Greece etc. In other words she is one of the cheapest heads of state in Europe and an elected head of state would probably cost more.
ONLY the Queen and her husband keep civil list money. All other members of the royal family are funded from the Queen's own private wealth or through revenue created by land they own. For example, Prince Charles is independently wealthy because he owns land that he makes a profit from every year (Duchy of Cornwall) and other members of the royal family have their costs reimbursed by the Queen every year from her own money. Most of the people who dislike the monarchy do so because they wrongly believe it to be an expensive institution. There is also the issue of the crown estates, which I won't go into, but suffice it to say we actually make a profit off the monarchy.