Comfy is not proper English, as it's informal and considered slang. The course is not allowing such slang words, but will allow colloquial English at times, depending on what it is.
With this, you're opening up occurrences where other informal words such as "cushy" is suggested for the same description.
While these words may be helpful for people to know who are taking English from Ukrainian, the course is designed to teach proper ways of speaking first.
No, as a native English speaker "convenient" and "comfortable" are two different things. "Convenient" is used to talk about something that is easy to get to or access. "Comfortable" is used to talk about something that is pleasant.
"Convenient footwear" may be used if the footwear is easy to access -- just like we have "convenience stores" that are plentiful and have gas, snacks, cigarettes, booze, etc. which make it "convenient" because you don't have to go to a lot of different places to get all those things. "Comfortable footwear" on the other hand would be footwear that feels nice to wear. Is there a similar distinction between "convenient" and "comfortable" in Ukrainian, I wonder?
Also, just a side note, "shoe" as a general word for the things that go on the outside of your feet is way more commonly used than "footwear".