1--You aren't giving up rights when you post something on your own website, you are using up rights.Basically, copyright is the right to copy. As the creator of a story, you own the right to make copies of it--no one else can do that without your permission.
So, you have the right to copy your story onto your website. The problem comes when you want to sell (actually it's more like renting or leasing) the right to copy your story to someone else.
Most people who might want the right to copy your story in their publication want to be the first one to copy it. It isn't very often that someone wants to copy something after someone else has already done it.
That's why magazine publishers call the rights they pay for "First serial rights" meaning the right to be the first serial (magazine) to publish a copy of your story.
If you copy your story onto your own website, then, you have used up the right to be the first one to copy it. (You didn't give up that right, you used it up.)
2--If someone has given you money so that they can be the first one to publish a copy of your story, then part of that agreement (or contract) you make with them may involve where else your work can appear while they are using the right they have paid for.
Many contracts talk about exclusive rights (and if they do, make sure there is a time limit on that exclusivity). That means that no one else, not even you, can make copies of your story while they have the right to make copies of it.
If you want to put your story or a part of it on your website, you need to make sure it doesn't violate the contract you have with the publisher who has given you money for the right to copy it first.
Always read and be sure you understand what is in a contract you sign.
3--A first publishing is usually determined by the number of people likely to obtain and read a copy of your work. Since the web is open to the whole world, putting something on a website is definitely publishing it.
Making as many as ten copies for your reading group to read is not considered publishing.
Putting something in the high school literary magazine is considered small press publishing and is not likely to reach enough people to interfere with "first world rights" which a professional publisher is interested in.
Of course, if READER'S DIGEST decides to reprint something you had published in your high school literary magazine or newspaper, that counts as publishing.
4--What you pay for with regard to copyright is the registration. As has been said before, the copyright is already yours when you write a story, but to register it with the government, you have to pay (to cover their cost of record keeping and so on).
The main reason for registering a copyright is so that if someone ever does infringe on your copyright (steal your story, for example, though there are other ways to infringe on someone's copyright), the law already says what their punishment is. All you need to do is prove the infringement, and the court will hand down the punishment.
If your copyright has not been registered and your copyright is infringed on, then you have to prove how you've been damaged by the infringement, and the judge has to decide if the infringement has really been that damaging, and what the punishment should be and so on and so forth--it can get to be a real mess.
Usually, the contract with a publisher should say that the publisher will register the copyright in your name. If the contract doesn't say that, ask about it.
Most writers don't worry about registering copyrights unless something they've written is likely to be made into a movie or a tv show and become franchised (where they have toys and underwear and bed linen and t-shirts and other money-making souvenir-type stuff to sell based on the story). That's when you want to be sure you control all of the rights to copy your story and characters.
Until it starts looking like Hollywood is interested, registering a copyright may not be that big a concern.
They aren't stupid questions, but they are asked a lot, and some people get tired of answering them over and over again.
I hope this helps.