>> Is it stable yet?
Been going strong for me for months now. I have around 20-30 Servers running in Hyper-V without issues.
>> If I installed Hyper-V on a new physical computer, would I be able to take the image of the servers I'm running on there and copy/paste it over to the new machine and run it there?
Yes, you can do that or backup the VMs with Backup Exec 12.5 or use SCVMM to migrate them between physical Hyper-V hosts
>> Is there anything else I should know about it?
There really is no reason not to do it.