In the US, the only job listings are in Alexandria, VA.

A patent form is rather complicated and the Patent office will charge you a fee to process. The minimum for a patent fee is 330$. So you should make sure that you do it correctly.

A published patent refers to the publication of a patent application, which is be reviewed by the Patent Authority (USPTO, EPO, JPTO, etc.). Once the merits of the application are acknowledged by the Examiner, process which takes from about to 2 to 4 years, the patent is granted (issued). The difference is that a patent application does not protect the inventor from any potential infraction to his/her inventive matter, and the inventor is allowed to start civil actions against the infractor, only once the patent is granted.

You'll need to do a prior art search to make sure your idea doesn't already exist. If it doesn't, you'll need to send design information to the USPTO where a patent examiner will check it. To expidite the process, you should hire a patent lawyer.

It depends on the type of IP you want to work in. Copyright and trademark tend to be oriented toward the humanities, but someone with a math degree would fit in as a patent examiner.