Here is what I know about DNS ('open' DNS is a new one on me):
DNS (Domain Name Service (or Server or System)) is a computer that has the DNS service installed on it. Sometimes there are public DNS servers (these respond to Internet requests like when you type http://www.microsoft.com into your browser) and sometimes there are private DNS servers (these respond to internal requests for resources such as your office's private server or your networked printer).
DNS servers work like this:
Every website and place that you go on the Internet has an IP address. When you type in a http://www.microsoft.com (or any other site) into your browser, the browser sends a request to a public DNS server (if searching on the public Internet) which takes 'www.microsoft.com' that you typed in your browser and resolves the plain English name of 'www.microsoft.com' into an IP address so that the DNS server knows where to send your browser to get a web page to display. The DNS server is said to resolve the friendly name to the IP address for Microsoft.
To answer your questions now,
1. I've not heard specifically about 'open' DNS, but DNS is very much utilized and required for the entire Internet to work properly. You could get to absolutly NO website (or any other http resource) without DNS servers unless you knew the exact IP address of every site that you wanted to visit.
2. Already answered what DNS is, I hope.
3. You can certainly use a private DNS server internally. In fact in most all modern networks past Windows NT4, you are required to have one to operate correctly. Typically, your router provides this function on a small home network or other protocols are used like WINS (similar to DNS but ancient.)
4. DHCP servers assign IP addresses and are used in conjunction with DNS servers. DHCP updates DNS with IP address information to keep names and IP addresses updated constantly.
5. You can use either a single DNS server or a primary and a backup (secondary). If a small network, you can get away with one with no problem (in fact, you can use the oldest computer you have to handle the DNS portion as it's not very taxing on the hardware at all.)
6. Depending on your network's construction and size, DNS servers are usually the best way to go with more than a few computers or when you have an Active Directory domain running with Windows Servers.