Adam,
If I didn't misunderstand your current versions....
Question 1 - Can I use the existing VirtualCenter server and add the new hosts to that, or do I need to install a new VCenter server? You will need to upgrade or install a new vCenter (if it were me, I would start from scratch, the upgrade process from 2.5 to 5.1 will take you a while IF there is even a supported upgrade path. The DB will need to be upgraded cleanly as well.) since the 2.5 vCenter will not support the host versions you are talking about. Just start from scratch if that's an option (purely my personal opinion), based on the assumed size of the environment, it shouldn't be too difficult to configure everything. Also, since you are jumping so many versions, you have to account for things like schema changes to the DB, new requirements (i.e. SSO) for vSphere, installed agents/tools, and the amound of updates involved. Also, Win Server 2008 wasn't around when you (I assume) built this environment.
Starting net new on the vSphere components (hosts, DB's, vCenter, etc..) will ensure a clean start and future for the environment...just make sure to upgrade the tools and hardware for all the guests.
Question 2 - Can I Vmotion the VM's from the old hosts to the new ones? If not, can I simply import the VMware files on the new hosts? I would say unregister the VM's and re-register them in the new environment, taking care to upgrade your vmtools and virtual machine hardware very soon.
Question 3 - I know there is backward compatibility issues, but moving forward so many versions also concerns me. Will the VM's run on the new hosts? I would look through the compatibility guide prior to making any decisions...I haven't looked through it lately, but I am pretty sure you can....just make sure everything get upgraded accordingly.
Question 4 - Is there anything I'm missing? I'm sure there is....just depends on the environment....make sure you consult the compatibility guide before moving on any of this.