Like all German simple vowels, ö has two pronunciations: a “short” one and a “long” one. They are usually similar but (except for the pronunciation of a or ä) not the same.
“Short ö” is /œ/ while “long ö” is /ø/, if you can read the International Phonetic Alphabet. Or if you speak French, they’re the vowels of neuf and deux, respectively.
General rules, yes, but not enough to know for sure how to spell something.
A consonant may be written doubled (pp, tt, ck, bb, dd, gg, ss, mm, nn) to show that the vowel before it is short, e.g. raten ("to guess"; long A) vs. Ratten ("rats"; short A).
A vowel may be written doubled (aa, ee, oo) to show that it is long. (Instead of ii, we write ie.)
A vowel may be followed by h to show that it is long: ah äh eh ih oh öh uh üh. You can even find ieh, which marks the i as long twice.
But then you have, for example, Bett (which is definitely short: double consonant after it) and Beet (which is definitely long: double vowel) -- but Gebet also has a long E in the second syllable. Here, the marker of length is the single consonant after it. A hypothetical word Bet would be pronounced the same way as Beet.
Similarly, vage (vague) and Waage (scales, for measuring weight) sound exactly the same, but one marks the length implicitly through the single following consonant, the other explicitly with double aa.
And sometimes, the final consonant of a short word is not doubled after a short vowel. For example, Weg (way, path) has a long vowel but weg (gone) has a short vowel.
So there are general tendencies but you can't know for sure whether (say) long E will be written e (before single consonant) or ee or eh.
Sometimes, multiple spellings are possible -- as with Lehre (apprenticeship; doctrine) versus Leere (emptiness), which are pronounced identically (with long E in the first syllable) but spelled differently.
So: guidelines, yes, which can narrow down the range of possible spellings, but the exact spelling still has to be memorised.
Finally: short E and short Ä sound identical, so if you hear that sound, you can't know whether (say) Stengel or Stängel is correct. (In fact, the former was correct before 1996, the latter after 1996, for complicated reasons.) And sometimes the difference is used to indicate separate meanings: Lerche (lark) is a bird, Lärche (larch) a tree. And Seite is a side while Saite is a string on a musical instrument -- pronounced identically.
can someone explain to me how to know which nouns will use das, die, or der as the article?
German nouns have 1 of 3 genders: masculine, feminine and neuter.
German nouns have 1 of 4 cases: nominative, accusative, dative, genitive
das is "the" for neuter nominative nouns and neuter accusative nouns.
der is "the" for masculine nominative nouns, feminine date nouns and feminine genitive nouns.
den is "the" for masculine accusative nouns.
That doesn't explain at all what you need to understand the issue, but I hope it gives you enough of a lead to look up what you need to.
I know what you mean. Duolingo will frequently test without teaching. It turns out that research supports this approach: an answer that you got wrong, even if through no fault of your own, will be learned more securely.
It's an approach that annoys people, but it is effective if they stick with it.