They are not exactly the same. Also, النور (al-noor) means (the light).
Alkhair الخير (al-khayr is more proper) means "good" as oppose to "evil" but not "good" as an adjective. This word can come also in the context of "fortune" or whatever is owned (money and such) because some Arabs in ancient times (and til this very day) call such thing as "khayr" or (khair) because it is one of the goods of life.
Now, expressions like مساء الخير translate as (evening of goodness), and the reply to this is (typically) مساء النور (evening of light/bright evening). It is for wishing one an evening full of goodness and well-being, without any evil, and the reply is wishing a bright evening; essentially almost the same mutual meaning as being an evening full of good things and with no evil. Thus, as you can see, the Arabic phrase or expression is not translated as it is into English but rather to something equal to it in usage or meaning in general.
Jayyid جيّد is the direct translation for "good" as an adjective. I would say this adjective is used mainly for quality-describing, or qualitatively.
Payyiab: Well, Arabic has no "P". I think here you mean Tayyib طيّب. This word can have various meaning depending on the context. One of the most common meanings is: kind (as in kind person, nice person). Another meaning comes in the context of being good in taste or good in smell, even though there are words for such adjectives that are more precise, but it is possible to use it in these situations as well. This adjective is quite connected to "blessings" as well. So, in a sense it can mean "good" in English, but it is not used for "qualitative" purposes specifically. For example, I can't translate "good pen" as قلم طيّب; It kind of does not make sense here. But it should be قلم جيّد.
Morning: صباح (cabáH).
Evening: مساء (masá2).
As for the translating from Arabic into English, maybe Duo did a mistake here I'm not sure (I won't be surprised), but anyway, the phrases صباح النور and مساء النور are mostly used as a reply for صباح الخير and مساء الخير (good morning, good evening, respectively).
As I play the audio here (on PC) the name is spelled correctly as (sárah).
Anyway, if you are just pointing out to the letter at the end, this is ta-marbúta (tied Ta) which has two spellings; Either as a H or as T. Since the word comes at the end of the sentence and no need to continue the speech, the last vowels are usually dropped (except in few cases) and when Ta-Marbúta has no vowel (or as we say "stable") then it keeps its sound as (H). This is regardless of the case.
Otherwise, if the word comes in the middle or in the beginning of the sentence and under whatever case influence, then we need to add vowels to the ending of the word, and when the word ends in Ta-Marbúta we have to change it from H to T.
In a nutshell, Ta-Marbúta (ة or ـة) is a single letter glyph combining two sounds altogether.
السلام عليكم يا تج_ق٨
(1) I am wondering why native Arabic speakers, including you, never explain them about the ending sound of the word after yaa. As Suxanxie says "if it's case ending, wouldn't it be nominative case here?", she asks about the status of the ending sound for سارة. But, I see native Arabic speakers just say they drop the ending sound. So, why is that, TJ_Q8? I am not asking you about the ending sound (because I know how the ending sound is in this case), but the reason why people don't answer it directly.
(2) So, once I asked the ending sound in some cases to some native Arabic speakers, they didn't know and offered me an Arabic teacher's number to discuss it further. I was a bit surprised, what happened there?
جزاكم الله خيرا.
Dropping the last vowel (the declension vowel) is done at stops, or at the end of the sentences. Speaking of the Standard Arabic of course. When you read Quran for example and you reach the end of the Aayah, you would usually stop without spelling the last vowel. Unless you want to read continuously, then you would pronounce the last vowel in the last word in the Aayah.
As for Arabs now, their language(s) is different than the standard. There are various dialects and these dialects are affected by many other languages. In dialects, the importance of grammar is lesser little bit - and because of that the ending of words (declension) is not quite important in dialects. Also, in dialects, many words did change in meaning from the original meaning, and spelling some had changed as well. Linguistically speaking, dialects can be considered languages on their own in the same way is Spanish, French, Italian, Portuguese and many others in relation to Latin. Thus, even though we study Arabic in school, but not all people are good students you know (I myself was a failure in Arabic classes back in school days). To make things worse, new generations do not have that interest in their own mother tongue and see it as something "old fashioned" or "retarded" to say the least. Because they lost faith in their own culture as it is with all the global issues and modern life requirements.
(1) "Why is there a "-ta" sound at the end of Sarah?"
If you have really heard "yaa saara(ta)", it is grammatically incorrect!
(2) "if it's case ending, wouldn't it be nominative case here?"
After the vocative particle, it would be in the accusative case (لان المنادى من باب المنصوبات). However, in the "Sarah" case, the ending sound is not -ta (like you have heard). But, it should be -tu, ie. "yaa saara(tu).
Hope it is clear now.