PPRuNe Forums - Questions regarding UK-USA training/visa

I have been trying to search for the answers to these questions but have had no luck. If anyone could give me some definite answers it would be very much appreciated.

1. A visa/TSA clearance is required for UK citizens to undergo flight training in the USA. Now, does that mean that I can have flight training but just cannot log the hours (or the hours would not count towards certification), or can I not legally fly an aeroplane under instruction full stop, regardless of logging hours or not?

2. What is the deal with employment in the USA for UK pilots? If one undertook full FAA training and wanted to then work as instructor with that school, would it be through an H1b (or similar) visa, or is there an alternative? I ask, because according to my research, the main factor preventing UK citizens easily obtaining US work visas (in any industry) is the clause that states that the employer must have first advertised and been unsuccessful in hiring a US citizen for the available position. Obviously, there is no way a US flight school could truthfully say that they couldnt find a US instructor to do the job.

Apologies for the seemingly generic questions, however I could not find any up-to date information on here.

Thanks

PP