How and when is Android's default browser updated? I am used to browsers having exploits found on them being fixed and having new versions of the browser with the fixes.

In the Android native browser it seems there is no such process.

Does this mean that exploits found for the native default Android browser are never fixed until you upgrade your ROM?

Is there a version history for the native browser?

When I look at my user agent info for my Froyo 2.2 Galaxy S I9000 I see Gecko version 533.1, which is very old. Does it mean exploits exist for this and there is no way to patch them?
You are correct. The built-in browser is only updated when a full new ROM is installed. If there is an exploit, the only thing to do is wait for a new version of the OS to be released. Alternatively, you can use a third-party browser such as Opera or Google Chrome. Those are installed as apps and can receive updates independently. 

As far as I know, there is no documented version history for the native browser. The best you could do is look at the GIT revision history for the browser.