Why Да here is "Yup" and not "Yes"? Also, are there different ways to say "yes" in Russian, just like there are "yeah", "yep", etc. in English?
And why should one say 'Да, доброе утро'? Is this a normal way of saying good morning?
I am confused about this too. Is it sort of like saying "Good morning to you too"?
Yes, it is. утро is neuter, день and вечер are both masculine. The feminine version of добрый is добрая.
Is добрый the 'regular' version? E.g. If I simply wanted to say "Good.", I'd have to say "добрый", and not any other version of the word?
добрый и хороший --these are two different words, but they are very similar in meaning . You can wish Good day and you can say about a human that he is a good (хороший) person- it's mean that he is good in general, but you can say that he is a kind ( good-добрый) person -it's mean he is not angry man . For example-- When I'm not hungry I'm a kind person --- Когда я не голоден я ДОБРЫЙ. In Russian Доброе утро, Добрый день, Добрый вечер, Доброго времени суток, Доброй ночи ---it's as a greeting but more formally, it's established expression. Добрый we use in established expressions and quality to humans and animals. Хороший we use as to everyone as to everything--- Хороший человек. Хороший день. Хороший стул. Ты сделал свою работу хорошо. And "хорошо" you can say as agree ( Купи молоко!--Хорошо. (Ок) or action approval Sorry for my English I started to study it not long ago)
I believe one would use "хорошо" for "good" in that manner, though I am not a native speaker. My Russian teacher would use "хорошо" as a compliment when we did something well. добрый I think translates more directly as "kind" or "nice". Other than these greetings, it is used to describe people ("He's a good/nice guy"). Maybe a native speaker on here can help clarify?
Russian dictionaries present adjectives in their masculine form, so you would see "добрый" listed in bold.
In my experience, "Добрый" is a common response to "Добрый день" or "Добрый вечер".
I used to greet my young children in the morning with "Добрая утка!"
Why is it «добрый день» but «доброе утро»?
 Ah, it has to do with the gender of the noun... :D
Can I just say доброе ночи for good night the way its used for good morning (доброе утро) instead of saying спокоинои ночи?
Yes, you can say "доброй ночи" (just not "доброе ночи", "доброе" is neutral and "ночь" is feminine).
Why not добрая ночь? Apparently Доброе утро and добрый день are both nominatives or accusatives, but спокойной ночх seems to be a genitive. Why the different case ?
i typed "да, доброе утро" and still rejected, the correct answer have no difference from what i typed this must be a bug
Imagine it being said by the Swedish Chef, and it makes more sense.
Русский: "Доброе утро, друг!" Sweed speaking bad Russian: Утро да, доброе утро!