You need to memorise the gender of each noun. Memorise "der Käse = the cheese", not "Käse = cheese". There's no way around this. There are some hints, but they only cover a fraction of all words and they're not 100% reliable.
Biological gender and grammatical gender are not the same. In general, grammatical gender is arbitrary. http://en.wikipedia.org/wiki/Grammatical_gender#Grammatical_vs._natural_gender
I noticed that sometimes the masculine article 'Der' is pronounced "Di-er" (example: der Käse, der Mann), while other times its pronnounced as a short "Der" (example: der Apfel). Is there some rule for this (Before certain letters/words or before words of a certain length), or am I just being over-sensitive to Dualingo's recordings of the sentences?
That would mean that both men and women eat bread (das Brot) but men only eat cheese on bread (der Käse) and women only sausage (die Wurst), never vice versa or together?
I'm afraid there isn't really any logic to grammatical gender in German -- it's just something you have to learn.
(Also, nouns are always capitalised in German: das Essen, die Schokolade and not das essen, die schokolade.)
Who uses a thing, or who needs a thing, has absolutely nothing to do with the gender of the noun. As mizinamo explained abive, you just have to memorize the grammatical gender of each noun as you learn it. Suggest using Quizlet.com to practice with flashcards (either online or you can print them out). Here's a great set: https://quizlet.com/389861116/german-food-flash-cards/
They don't sound the same (although they're close). "a" and "ä" are two entirely different sounds. "a" is an open a, like in 'apple' while "ä" is with a half closed mouth, kind of like 'e' in melody.
I'm not sure I understand your question about spelling the word, but if you are asking how to type it:
On a smartphone - a long tuch on a vowel's 'key' on the keyboard opens a sub-list of alternatives for the vowel with umlauts (relevant not just for german, but also to other languages using the latin ABC). same applies for a long touch on the 's' key, opening a sub-folder with the option to pick the 'ß' character from it.
on a computer - there are small buttons under any place in duolinge where you need to type German, with the vowels including the umlauts and the 'ß' character. just click them when you need them...
"well" is usually an adverb -- you can "sing well", for example (= in a manner which is good).
But "to be" joins a subject to a predicate (a noun or an adjective); you don't usually use an adverb with it (it would mean something like "I exist in a way which is good").
"well" as an adjective means something like "healthy" -- e.g. "a well-woman checkup" or "I am not feeling well".
But we are not talking about the cheese's health, so the adjective "good" is correct here.
I guess the same way as you would learn the gender of all other nouns. I prefer using actual flashcards myself, old-fashioned though it is, with pictures and the nouns with their correct article: der Apfel, die Kartoffel, die Erdbeere, das Brot, and so on. Quizlet has many flashcard sets to use. Here's one: https://quizlet.com/389861116/german-food-flash-cards/.
Das is the article for neuter nouns: das Buch, das Auto, das Haus.
Der is the article for masculine nouns if they are the subject of a verb (nominative case). Examples: der Käse, der Wagen, der Tisch.
Den is the article for masculine nouns if they are the object of a verb (accusative case). Examples: Ich sehe den Wagen. Er kauft den Tisch. Wir essen den Käse.