Yes. Mtanzania can either be the adjective "Tanzanian", or the noun "a/the Tanzanian". In both cases it refers to a person, not a name, because of the M-prefix. I'm not sure excactly what "Juma is a Tanzanian name" would translate to, but it might be "Juma ni jina Tanzania"
It's an accepted translation now. It might not be a perfect word for word translation, but "the name of a tanzanian" is hardly proper english, so "a tanzanian name" makes a lot more sense, especially if "la tanzania" is how that idea would be expressed in Swahili.
They all mean "of". The word for "of" is essentially just -a and it takes a consonant prefix which depends on the class of the preceding word.
There's a table here.
It looks confusing at the beginning, but trust me, it gets easier as you go along. The noun classes are usually easy to recognise.
The main ones that can trip you up are the ones where the noun doesn't have a prefix ... the Ma-class in singular (such as jina) and the N-class in singular and plural (such as habari). Sometimes a noun looks like it has the prefix of one class but it's just coincidence and it actually has no prefix. I write down vocabulary like this.
jina la = name of
majina ya = names of
simba ya = lion of
simba za = lions of
mume wa = husband of waume wa = husbands of
All of the words for "my", "your" etc., will begin with the same letter as the word for "of" and they're also repeated in verb prefixes (with a few phonetic alterations).
Your sentence implies that the country Tanzania is also calles Juma, not that a/the Tanzanian person is called Juma.
It keeps saying "you need the article "a" but the "a" is not aption... Its not there...
Is there a word for "the" in Swahili? Or is it just given in the Mtanzania?
I believe this is the literal translation:
"Juma ni jina la Mtanzania" "Juma is name of [person] Tanzania(n)"
The M- before Tanzania means it's referring to a person, not an object.