Can I update Mail.app to fix persistent crashing?
The version of Mail.app available to you to is pinned to the version of OS X you're running as it is considered a core application that is bundled, shipped and installed with the operating system.

If you want a new version of Mail.app you'll need update OS X. Your iMac, assuming it's the Late 2006 model I think it is based on what you wrote, should be able to run OS X 10.7.x Lion. Upgrading to Lion is a fairly painless activity though you should be certain to follow best-practices and make a backup of your system before you do the upgrade. The upgrade tools will maintain all your Mail.app settings and you won't lose email, contacts or any other such thing like that.

Alternatively you can look for a third party mail application that's compatible with OS X 10.6.8 and try running one of those. Honestly, I can't recall any that were 10.6.8 compatible off the top of my head though.

A third option is to try and figure out what's keeping Mail.app from running on your machine. The best place to start is by looking in Console.app -- run Console.app and then start Mail.app and look at the error messages that are printed in the Console. They might shed some light on what's going wrong. Or they might form the basis of a new question here on Ask Different to figure out what's going wrong!