What do we know about the extent and limitations of their
Not a great deal, I'm afraid. Historically, Zygon technology enhances their inherent shape-shifting abilities and gives them the power to turn themselves into replicas whenever it is necessary according to Broton ("Terror of the Zygons", Series 13 episode 1, four parter). In fact, it is necesary to activate a body print every few hours or the original pattern dies and cannot be used again. According to TARDIS WIKIA, the likeness "wore off every two hours, necessitating its re-assumption" although I can't find where they got this snippet of information from.
It is speculated that, whilst the body-print technology allows near perfect replication of body, fabrics and materials, no item of clothing could actually be removed in the same way that a Zygon could not remove an appendage, sucker or a flap of skin. This would allow Zygon physiology to imitate clothing that could flap open such as a coat or scarf or even hair bands, scrunches or glasses (other then being rigid, a spectacle lens is not much different to an eye lens or cornea) but such items couldn't actually be removed. It would follow then, that an inhaler which is an item that is not worn and shouldn't be permanently attached to a hand as it would be a dead give-away in any given length of time. It is also reasonable to assume that the Zygon Body-Print Technology is smart enough to make logical decisions based on item function, composition or use and decide which items to replicate or leave out of the print. Items with chemical components and complex moving parts would be likely candidates for exclusion.
I would be very interested to know whether they can also replicate natural body functions and odours such as sweat and bad breath or even fabricated odours such as perfume, hairspray or mouthwash. This would give Zygon enemies the opportunity to identify replicants of familiar persons by they're lack of familiar smell. It may look like the Queen and it may talk like the Queen but if it smells like Asparagus in an old sock...well then.