For professional drivers, how fast do they usually drive? :-)

The answer is, of course, "it depends." If you're a taxicab driver trying to get through downtown Manhattan, the answer may be "5 mph." If you're a NASCAR race driver, probably something higher.

For fast-paced first-person shooters, very high tick rates (so, low update times) are common on dedicated hosted servers. When I played CounterStrike, servers advertised tick rates of 60 and even 100 ticks per second.

For console-based games, 10 Hz is pretty common, because of the bandwidth restrictions.

Then there's the question of simulation tick rate (how fast do you step physics?) versus network send time (how often do you send packets?) which do not have to be the same.

In most professional games how often do servers update (not as in patches but other player positions and stuff) their clients or do they use a different method to sending out an update every so often? At the moment my server sends out sixty times a second and when the client receives an update it sends an update to the server. It works, but is this the traditional way of doing things?

Pretty much, but also depends on your bandwidth budget. The faster you send, the smaller the packets (less difference between updates), the more responsive, but the bigger the packet overhead becomes (UDP / TCP/IP headers).

As hplus says, usually, between 10-15 fps (console peer to peer gaming, where uploads are restricted), to 30 fps or more for dedicated servers (geared towards services many clients, with massive upload capabilities). Note that slower updates introduce more latency. Particular entity updates also scale depending on the importance from a given client perspective. Say if an object if at 3 meters away from A, 300 meters away from B, and you base your metrics to purely the distance from an observer, the object will send updates more often to A than to B, as B cares less about its actual state.

I had the Ultima Online servers running at 4 fps and SWG servers running at about 10 fps. League of Legends servers run as fast as they can in order to be as responsive as possible.