1. You can use AJAX to trigger the URL without redirecting the user to another page like explained on that post:
www.hikashop.com/support/forum/4-how-to/...e.html?lang=en#11987
2. You need to add an id for each Buy link like this: <a href"..." id="seat_X">Buy</a>, where X is the seat number. Then, before the 'return false;' code of the link, you can add the code:
document.getElementById('ID').style.display = 'none';
You need to replace ID by the id of your link.
Please note that 2 just hide the buy link on the current page. If the page is refreshed, or if another user sees it, the seat will still be available.
If you want to have a reservation system, that's way more difficult and you definitely can't do that in a joomla article. You will have to build your own small reservation component for that with a place where to store the reservations and then build the listing of seats dynamically from that data.