I don't know when to put "nzuri" and "mzuri". Don't the both of them mean "good" or fine.
It changes according to the noun class. "Nchi" is an n-class, so "nzuri". In contrast, an m-class requires "mzuri" (e.g. "mtoto mzuri").
To answer your second question, yes, they both mean "good". Nzuri and mzuri are the same word ("-zuri").
Why isn't 'Tanzania is a fine country' accepted? nzuri is also 'good,' right?
Tanzania is a great country. It's also a good country, a beautiful country, a fine country, an awesome country and a nice country. Just like with English and a multitude of other languages, one can use a more general term like "good" to embrace all these differentiated forms of "good" with finer nuances. But, on the other hand, you might like to refine your concept of "good" or "nice" by choosing a word with a more specific meaning or nuance. So DL is here trying to teach us the more general word, "-zuri", meaning "good" or "nice", both of which have very broad meanings. That's most important in the beginning for learning the language on a basic level.