The company you buy the domain name through determines the cost of the domain name, so it is always best to compare prices. Usually (dot)com domain names are more expensive then (dot)net, or another ending.
Yes, cheap domain names are secure depending on what company you purchase from. They offer low prices and the same reliability you would find with any other provider.
Cheap domain names can be found at Godaddy or Register. There are many domain sites you just have to find the ones that are good for you. Some are even free.
Dyn-DNS is free just for the domain name, you have other options available on the site for paid custom domain names. Go-daddy is really good and very popular. Check around online for other options...
You can look up domain names through many search means or by typing them into your web address bar. You can also try Bustamove which allows you to search for domain names to see what is available.
Yes you do, most web domain registrars give you all the tools you will need to build your website. You will need your domain name because it is the address where you can be found. A good one would be...