What Christian said. Grammatical gender is arbitrary. In this particular example there may be an explanation, however: The word is rooted in Latin 'caseus' with 'us' being a masculine ending. So, now we're stuck with the question why the Romans chose cheese to be masculine ;-)
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
As far as I can see, grammatical gender is called gender since the words for man and woman refers to different genders, and tend to be different grammatically too, like in German. So since it's der Mann and die Frau, "der" is considered masculine and "die" is considered feminine, and "das" is what's left over. But not only these words, but other terms for humans tend to also match these genders: der Junge (boy), der Onkel (uncle), der Vater (father) and die Magd (girl, now archaic), die Tante (aunt), die Mutter (mother). But it doesn't mean it's a perfect match, but tend to be a close one.
Then all other nouns just seemingly randomly fall into der/die/das and are therefore called masculine/feminine/neither, but there's nothing masculine about cheese, it just behaves the same as the actual masculine nouns for people.
But since the diminutive is neuter, and the modern term (or how modern things from 1750s are) for girl is "Mädchen", it's one of the odd ones. But if you do look back in the time, the old term was Magd which was feminine.
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?
On mobile, you can enable a German keyboard in addition to your other keyboards. Duolingo will ensure it's automatically selected for when you type German and makes Ä Ö Ü available, and ß will be through long-press on S if you can't get it normally.
On computer, you can enable the German layout in addition to your normal one. On Windows you can easily switch to it by Win+Space. You type Ü to the right of P, and Ö Ä to the right of L, and ß is located to the right of 0. Comma and period are located in the same position as most layouts. If you don't want to switch back and forth between German and your default to get the apostrophe; it's located to the right of Ä (ISO, European), or above Enter (ANSI, American), depending on your physical keyboard, and you need to press shift otherwise you get #. You only need this enabled while on Duolingo when typing, since you don't need to type ( ) ? + " / and other stuff.
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.)
Words come in three groups; der, die, das, and that's basically it as far as the language is concerned. There's no deeper meaning to it.
'der' is used for masculine nouns like Vater (father), Junge (boy), Onkel (uncle) and 'die' is used for feminine nouns like Mutter (mother), Magd (girl), Tante (aunt). And 'das' isn't used with people (outside of diminutive).
However, since 'das' is used for diminutive nouns (-chen), and since about 1750s, the preferred term for girls is the diminutive of Magd, this made girl to be 'das Mädchen', and appears as the outlier. But it still makes sense when you know the history.
But there's really nothing deep to it. They could just as simply have called them the derative, dieative and dasative :)
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.