In the United States, if I own land that has an area suitable for landing a small general aviation airplane, can I just create a private runway, or do I have to officially be registered with the FAA or some government agency?
In short: Yes, you can. The FAA doesn't really care, as long as you're not going to interrupt class B or C operations. And honestly as long as you're not hosting a dozen other aircraft, how are they going to tell the difference between a field and an airstrip? If you want it on a chart or you are near special airspace, you need to contact the FSDO and talk to them about it (it's good to do that anyway). They will send someone out to survey it and talk with you, and hopefully approve it.
It's the local authorities that usually ends up being troublesome. Your state/county/city might have laws that apply, and you'll need to find that out on your own. A friend of mine spent just a few days talking/meeting with the FAA on his strip, but spent weeks dealing with the state transportation department and a local ban on new grass strips which he sat right on the border of (as in it was a ban in a 50 mile radius and his house was on mile 50).
Persistence is key, good luck!
I have read of another who went to the trouble of registering their strip - worked out well when a power company wanted to run high voltage lines right over the middle of the strip. Since it was a registered airport, they had to run the lines elsewhere. Here it is, the 2nd answer: Why do farmers register their fields as airports?