Well, Latin do have some definite-article-like words also and they are : ille, illa, illud. I don't think that not placing any article-like words in Latin will make an equivalence of definite article in English, because such system of article doesn't exist in Latin. If you use "una femina" it will have a meaning like "that one woman alone" or "only one woman" (emphasizing on being alone) rather than "an unspecific woman" like in the meaning of "a woman" in English.
If you're using a mobile device, then the next time this happens try turning it 90 degrees and see if more tiles appear in the word bank.
Also, take a screen shot and file a bug report: https://support.duolingo.com/hc/en-us/sections/200864570-Reporting-Issues
Yes, that is right, and especially about eager, thirsty, hungering seeking. In this context, it means the seeking of knowledge.
A lot of verbs do this sort of thing, not only in classical Latin, but also in other languages. We have some of the consequences in English, e.g. "credit" and "credible" come from the Latin verb meaning to trust and to believe. Financial "credit" is about trusting that the borrower will pay the lender back, though mostly we have forgotten that and think of "credit" being the money.
There is another knowledge-seeking word, quaere, that's got a different nuance, more about getting specific answers. It is where "question" and "inquire" come from.
No one here can do anything about that. You'll need to post in the troubleshooting forum:
Or submit a bug report:
Duolingo Latin doesn't do macrons at all. Some of us find that frustrating because we find macrons helpful to learn, and they help to overcome difficulties hearing the recordings. Others say that Latin hasn't used macrons anyway except in very modern beginner's textbooks (and numerals). Modern Latin also used both ´ and ` diacritics, and no one seems to argue for that. No one seems especially bothered by the u,v and i,j usage either, nr upper case for proper nouns and the beginnings of sentences, nor 'traditional' punctuation.
If you are wanting to know the vowel length: this conjugation has a short e in the 1st and 3rd person, and a long ē in the 2nd person:
I/we study: studeo studemus you study: studēs studētis he studies: studet student
The textbooks typically give monēo (I warn) as a model.