what's involved in getting it under EU registry?
That will vary between countries.
I suppose VAT, anything else?
If you are going to fly any plane around Europe, you had better make sure anyway the VAT has been paid _and_ carry proof. Whatever the registration.
How is the VAT calculated?
That too will vary between countries. I seem to have heard that a catalog is used by German tax authorities, but have neither certainty nor details.