I guess there’s no distinct English word for "drogerie", but the Wiktionary states that "drogerie" is NOT the same as pharmacy (for which it gives the word "lékárna"). If that’s true, I think it might be a good idea to highlight it in Tips and Notes.
EDIT: I haven't tried "chemist's" as a reverse translation - would it work?
yes, Chemist's and drugstore would work as well. The problem is that, specially in the US (CVS, Wallgreens, etc. ) are a combination of everything. It really is a convenient store with a pharmacy in the back. This is not the case in Czech republic. Pharmacy (Lékarna) is strictly a place where you get both prescription and over the counter drugs. And you cannot really get them anywhere else. Not even aspirin or ibuprofen or cough syrup. In the U.S. you get all of the over the counter at every gas station. So it is difficult to align a proper translation for two vastly different places.
Drogerie in Czech is a place where you get all cosmetics, all cleaning products, toothpaste, shampoo, diapers but no drugs.
Absolutely, if it's possible. I'd really use "pharmacy" - "lékárna" and "drugstore" for "drogerie". (Even drugstore wouldn't be exactly the same a drogerie but it would be much closer). In fact, the things you may find in most Czech "drogerie" that have somehow to do with health would be things like sun protection or plasters (band-aid). But, as mentioned above, no drugs as such (except in a mini-pharmacy as a distinct shop on the drugstore's presincts.
This is a very unfortunate translation. No, sorry, this is an utterly incorrect translation. I guess this is the way duolingo works: you can translate a drug store or chemist's as "lékárna", so it will work also the other way round. However, it should be crossed out. This might be even quite dangerous / painful for an expat to learn that he can ask for a direction to a pharmacy by asking "kde je drogerie". I have heard countless stories of expats trying to have a conversation with an MD in English, and it's already painful enough without making it harder for them to find a pharmacy.