For some a proxy server is used to get you into blocked sites.
Sometimes for unknown reasons an ISP will block a site. Or they will block chat lines as well.
Right now ATT is blocking a common info site. This has been going on for months and is nation wide. I use a proxy to get to that site daily. Verizon is blocking chat lines such as MIRC for some unknown reason. Calls to them only results in frustration because they say they are not blocking. A ping to an Ip address shows the node that is blocking.
This is partially true. MIRC is a program, IRC is a protocol. A proxy is a good way to get around upstream controls, however... that's not what he's trying to do.
A proxy server will get you a static IP address. Users of cable who cannot get or do not wish to pay for a static IP address will use a proxy server to get a static IP address.
A proxy will NOT 'get' you a static IP address, however will make it appear that you are coming from another address each time you use the same proxy, provided addresses aren't utilized in a load balancing or round-robin configuration. You cannot accept inbound requests via a proxy IP, nor can you establish any ip-based security as untold amounts of other users are using the same IP as a proxy. (This excludes paid-for access, in which you can pay to have a dedicated proxy ip just for you- still won't support inbound requests and is just as insecure as using your own IP- moot point)
Other use a proxy service to filter incoming data. I set up one for a company that did not want any porn coming into their system.
This is true, but not his goal- and generally is not implemented on free/cheap services.
Others wil use them for speed gains. To speed up access to resources (using caching).
This is only true if you are running your own proxy (preferably locally) and have it configured to do so.
Some use them for log / audit usage to provide company employee Internet usage reports and to scan outbound content, e.g., for data leak protection.
See above, plus a few paid-for services. Also, doesn't appear to be the OP's focus.
Internet-facing proxy is a front-end to control and protect access to a server on a private network, commonly also performing tasks such as load-balancing, authentication, decryption or caching.
See above, again. There are a number of other more secure controls and I haven't seen this method implemented in years.
To dismiss a proxy as a mere hacker device is a little narrow in scope. Looking at cookies is a long dead way to track who you are. Your configuration via your browser is a much better fingerprint. For example. You will have a font library and your browser will list every font there. When you visit a website that list will be revealed via your browser.
Font cache? Seriously? That can be mined, but is far from a security risk. Browser configuration? Yes, me and the other 100,000 people that have the same browser and OS.
Some companies more often than not use them to eavesdrop on the employees. So watch it employees.
100% true, but not applicable.
(edit) I had to add point by point... let's be honest here... no need to inundate people with useless information and misinformation... why not stick to the question?