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"
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).