In this case, I believe the use of ya or za with Habari depends on the country. Ya /za should agree with the preceding noun in this case habari (news). Habari is an N class noun. Therefore, za should be used grammatically speaking since ya is for singular N class nouns and za is for plurals.
However, in Kenya, this rule is not followed. In Kenya, they say habari ya jioni, mchana, asubuhi, etc. I am told that in Tanzania the grammatically correct za is used. But, if you use za in Kenya they will understand what you mean. You will just sound like a tourist, which is fine.
Some people explain that za is used for general politeness and ya is used when you really want to know what's going on. See website below. http://www.learnswahili.net/swahilicourse/learnswahilicourse_lesson2.html
And, perhaps this is why even Duolingo uses ya and za at different times. What is the news of the day? (General) vs. What's your news of the day. (Personal)
So maybe Kenyans choose to be less formal and that's why it is commonly used in Kenya. I don't know. I just know Kenyans say Ya even though it should be za if you are following the rule.
There are many more than two words for "of". It depends on the class of the noun that goes before it. You can check out the table here: https://en.wiktionary.org/wiki/a#Swahili
Don't get overwhelmed ... each noun will tend to be surrounded by other words that show the class of the noun, so over time, I'm guessing we will get a feeling that habari goes with ya (singular) or za (plural) and never with wa or any of the other forms. It's kind of complex but it's not actually that crazy, just very different from European and Asian languages.
I think a literal translation is not needed for this greeting in the morning. What an English speaker is most likely to say is "Good morning" but could also say "How's the morning?" or How are you this morning?". If I say to someone "What's the news this morning?" I can expect an answer such as "There was a plane crash in Chile and no-one survived"