Can I Update my T-Mobile G1 (Also known as the HTC Dream) so it runs Android 2? Is it possible to do this without rooting the phone?
No. You must root your phone and install a mod such as CyanogenMod to get your phone running a semi-equivalent of Android 2.1+. There is no stock rom for Android 2.1+ for the G1. Additionally, if you do go with CyanogenMod, not all features are fully supported due to hardware limitations, such as LiveWallpapers.
You can run CyanogenMod 6.0, which is Android 2.2 aka FroYo.
A few tipps of mine: If you don't want it to be terrible slow, disable dithering and all effects in the settings. Maybe enable the just in time compiler (jit). With the app titanium backup you will be able to remove some "garbage" apps which come with the cm release which is also helpfull.