Well, in English I would say "spacious" is used for a sense of "wide" larger space to move around, and "big" is probably more general and more inclined to be used for sizes.
In Arabic it's almost the same. The adjective كبير is used for sizes, and ages as well, while واسع is used for something of much free space (as opposed to ضيق -DHayyiq: Narrow-)
No it's correct. It's just on Duolingo... the speech machine does not say things correctly and does not care about grammar. In other words, Duolingo is somehow based on a mix of Arabic (standard and dialects from here and there).
For example, your phrase here (big building) should be more precisely (bináyatun kabírah) or (bináyatun kabíratun) -to make it further complete- but as the speech machine put it here, it doesn't put the proper ending as it should be in standard Arabic, but I'd say this is how someone would say it in his or her own dialect (because in dialects people stopped caring for the proper ending/declension of words).
So, Duolingo might serve as a starting point, but definitely not the ultimate goal to learn Arabic (even if you are planning to learn a colloquial/dialect version and not the standard). The technical problems and the strange mix of sentences (and the lack of explanations) makes it more suitable for a "traveler's guide".
with proper Arabic, it should be (bináyatun kabírah) or (kabíratun) - this word is at the end so the vowel -un can be dropped.
Here we have a noun (feminine) which ends in Ta-Marbúta ة . This letter merges H and T in a single form. This is because in Arabic, there are some nouns that end in "H" usually, but when are moved with vowels and come in the beginning or in the middle of a sentence, eventually this "H" turns to "T" (just a property of some words and majorly feminine words). So, instead of changing the spelling of such words from H هـ to ت ... this letter, Ta'-Marbúta, was invented to merge the two into one.
The -un sound is called Nunation or Tanwin. It has many uses and signifies some aspects but for the time being ...... and in this instance, just imagine it is equivalent to the English a/an article (i.e. the indefinite article). Thus, a word in Arabic, if not defined with AL in the beginning, then it is marked as indefinite by this Tanwin at its end.