Dedicated servers are more secure than shared servers. On shared servers anyone is able to access the server, whereas with dedicated servers, only those who are authorized can access the server.
Generally speaking, no. Windows has a slower patch cycle than most Linux distributions, thus exploits are exposed for longer and have a greater probability of being hit.
In my personal opinion, No, a Linux server is the better answer for the web. I would recommend a windows server only when running asp/.net, and with the introduction of mono (an apache module), Linux...
Linux is better than Windows for servers because it is more stable and more secure which are two important reasons why it is better for your server and why you can trust it more.
A web server strictly speaking would be a server that hosts web sites. The choice you have least expensive first: Shared Hosting - a share on a web server, no fixed CPU, Disk Space, BandwidthVPS - A...