I recommend getting a reusable lint roller to more easily lift fur from surfaces on a routine basis: https://www.reddit.com/r/Rabbits/comments/pd9y9i/cleaning_reusable_lint_remover_to_reduce_fur/
Re: hay bits, where and how you present hay can impact how much will track everywhere. The larger the litterboxes, hypothetically the better hay is contained.
Re: the fur everywhere, your lionhead is likely molting right now so ideally it won’t be such high volume for too much longer.
Re: hay dust, if agreeable with your circumstance, you can always switch to a less dusty hay like orchard instead.
What will cause a respiratory infection is bacteria paired with a rabbit having a weakened immune system — I would not get concerned over the mere presence of dusty hay and fur causing a respiratory infection (especially when those are the two things arguably impossible to completely remove from the environment). Remember rabbits actively, throughout the day, plunge their head in hay to eat, going nose-first into it, and they also groom themselves daily, actively eating that fur.
The answers provided in this discussion are for general guideline purposes only. The information is not intended to diagnose or treat your pet. Seek the advice of your veterinarian or a qualified behaviorist.