1. It is possible to be both self-employed and employed but HMR&C will want to make sure that any "joint" costs are fairly proportioned between the two businesses.
2. Highly, highly unlikely. In fact, I would say "no" but I have heard of people who claim they have done it. I am sceptical. All I can say is that they were dealing with an Inspector who didn't know the rules and that is also unlikely.
The point with VAT is that it is a tax on the end user. If your customer is someone who can reclaim and pass on the VAT to an end user, then you can easily obtain VAT registration.
However, the self-employed route also has some income tax advantages especially when it comes to claim tax on job-related expenses and so HMR&C are quite strict on whom they regard as self-employed and whom they do not. Some cases will go to the Commissioners. However, I do understand that they "turn a blind eye" to flying instructors as it is appreciated that that is how the industry works.
Cheers
Whirls