Yes, that lens will work fine on a 5D (or any other full-frame Canon camera). The only Canon lenses which are incompatible with full-frame sensors are the EF-S range.

You will notice a change in zoom factor though. The 1.6x smaller sensor on your 450D means that the lens is currently giving you a zoom range equivalent to 112-480mm. On a full frame camera you'll notice it really is 70-300mm - things won't look quite as close when you zoom right in. However, the massive difference in resolution (12mp on the 450D vs. 22mp on the 5DmkIII) means you can crop in post-production to achieve the same effective zoom level and still end up with a much bigger image. You certainly won't be losing anything.

Yes, that lens will work fine on a 5D (or any other full-frame Canon camera). The only Canon lenses which are incompatible with full-frame sensors are the EF-S range.

You will notice a change in zoom factor though. The 1.6x smaller sensor on your 450D means that the lens is currently giving you a zoom range equivalent to 112-480mm. On a full frame camera you'll notice it really is 70-300mm - things won't look as close when you zoom in. However, the massive difference in resolution (12mp on the 450D vs. 22mp on the 5DmkIII) means you can (if you want) perform the 1.6x crop in post-production without losing too much image resolution.