1. No. The word "right" in this context is thrown around a lot these days, but I think it is worth noting the original list of rights in this nation included tools such as equal opportunity which gives the individual the means to go as far in life and be successful in pursuits of their choosing (and please note that success is different for different individuals) based on their willingness, ableness, and effort. This is a strong foundation for every individual in a nation to do their best and devote their resources to pursuits best suited for them. When you have a nation of individuals all working optimally for their own success, you have a successful nation whose curiosity and work ethics lead to a highly efficient, technologically advanced state. Nowadays, instead of people using their original rights to work towards fulfilling their needs and wants, they are declaring their needs and wants rights that should just be given. History has shown that such systems breed a population that is inefficient, stagnant, and technologically lagging. The evidence is stacked high as seen in the Soviet Union, North Korea, China, Vietnam, Cuba, Greece, and many others.
Now that said, birth control is an awesome thing to have, and it makes sense for any sexually active adult to invest in it and use it except when trying to have a child that they want to have and the means to raise that child to be an awesome adult.
2. So it is not a right, just like any form of healthcare shouldn't be. However, keep in mind that freedom of religion goes both ways. Just as it doesn't make sense to make organizations pay for all this stuff when they don't want to, it also doesn't make sense for religious organizations to reach out and try to dictate their misguided and harmful policies on everyone else, something that many religious organizations are constantly attempting to do. So they are not really innocent here. Many religious organizations don't just want to not pay for contraception, they want to ban people from its production, purchase, and use. So they will say their freedom of religion is being stomped on at one point, then smash everyone else's freedom of religion at another point. It's madness.
3. I'm not a fan of slippery slope arguments, but it certainly is a step in the wrong direction when people realize they can legislate the taking of resources from one group of people to give to another by majority votes. It may look shiny and wonderful for the majority, at least in the short term, but the long term damage will be unbearable.
4. Because it is no longer designed to proceed based on the best interests of the United States; it is designed to bring in a majority of votes.
A response to an individual statement:
"Why don't we just have socialized healthcare so that everyone has access to what they need..."
Because there isn't an infinite supply of every amazing cure we've ever made and new amazing cures yet to be invented don't appear out of thin air. This stuff is expensive. You can either let market forces determine how these resources are allocated, and this system is most efficient in maintaining supply and driving innovation to improve or create treatments. Or you can subsidize it, which creates an artificial distribution and these things have never been sustainable forever whenever it has been instigated.
"...employers can just deal with the fact that their employees' medical habits and needs are probably none of their business?"
If a company is paying for something... then it is their business. Their money is purchasing it. It's a real shame too because not only are you increasing their costs, but you are also diverting their attention from core business so you are inevitably damaging their capability to innovate and produce superior products. As a simplified example, let's consider Ford. The mission of Ford is to make money by producing outstanding, innovative, and reliable ground transportation vehicles. They are constantly working to make things that meet whatever mission. Gas prices went up? They work on gas mileage. The market for carrying heavy equipment heats up? They work to increase capacity or engine power. Anytime you introduce an element outside this core business, you are affecting the ability of the company to perform that core business well. In this case, you are making them pay for the employee healthcare. More cost. They need to manage the health plans. More overhead costs. Anything they can do to limit this impact would have to be considered to stay competitive, especially when some competitors are located in places where this mandate doesn't exist. This is the kind of thing for why my employer now does health screenings. They now have a real reason to be concerned on what my health is because it will affect their bottom line. People cry afoul of privacy being affected, but they should have known they were asking for this when they voted to make their healthcare their company's business.