Does flight training have any impact on my H1B visa?
Having flight training has no effect at all on your visa status. It's actually the other way around: your visa status can determine whether or not you can have flight training. But you can definitely get flight training with an H1B.

And flying has no special impact on your job: how you choose to spend your free time is none of your employer's business. There could be some very specific scenarios where flight training and work intersect, e.g. if your company is in the aviation sector you might be able to get them to pay for training. Or if you're a very senior executive at a very large company then the Board of Directors might have concerns about you spending your time on 'risky' activities. But otherwise, flying has nothing at all to do with your work.