It's not a cognate, if that's what you are claiming by "coincidence". Wiktionary has an article describing the origin of the word (if one should trust such sources). But it's popularity today is very likely due to similarity to the Indoeuropean words for name. According to the wiktionary article, the word appeared in writing a few hundred years ago, but became popular around the 1860's and 70's.
It's not a coincidence: "nombre" in Spanish, "nom" in French, "nome" in Italian. It all comes down from Latin which influenced all these countries even England. Alexander Magnus' empire reached many parts of the east of Europe reach in many parts of Asia so probably it's an influence from then.
No. The Persian nām, the Latin nomen, the English name, the Sanskrit nāma, and the Russian imja are indeed all from the same Proto-Indo-European root word. The Persian word is directly from PIE, it's not borrowed from Latin or anything.
The Japanese word is a coincidence, it does not derive from PIE or any of its daughter languages.
I hope you're prepared for another long answer ;)
Before I discuss each of the kanji you asked about, let me put out a disclaimer. While each kanji on their own will mean something that usually informs us about why the kanji combinations (i.e. compound kanji words made of 2 or more separate characters), you should remember that the actual meaning of the combined word is distinct and separate from the meanings of the individual kanji.
Ok, let me go through them in order:
・前（ゼン、まえ）is commonly used on its own as a preposition (both temporally and spatially), which can take many meanings depending on its use, including "in front (of)", "before", "ago", or "previous".
・名（メイ、ミョウ、な）on its own can mean "name", but I think you only really see that usage on forms and paperwork, as a shortened version of 名前.
It also carries the meaning of "reputation" or "distinguished". I'm definitely not an expert in linguistics, but the combination of 名前 could be referring to "the thing that precedes your reputation".
・学（ガク、まな.ぶ）is a kanji that is barely used on its own anymore, but it does appear in many kanji combinations. It means "learning", "study", or "science".
・生（セイ、ショウ、なま、う.まれる、い.かす、い.きる、は.やす、etc.）has a lot of different meanings, as you can tell from its many readings. On its own, it can mean "raw" or "genuine", but the kanji also carries the meanings "life" and "birth".
Putting this together with 学, i think the idea is of "someone who is in the studying part of their life". (Again, just my guess.)
・先（セン、さき、ま.ず）is confusingly similar to 前, in that both are commonly used as prepositions meaning "previous", but 先 also refers to "ahead" or "future". I believe that, in the cases where 先 means "previous", it is referring to "the thing that came ahead".
Putting this together with 生, I think it refers to "the person who has lived ahead of you", and presumably has more experience and knowledge which they can pass on to you.
Obviously, I'm not going into too much detail about what other possible meanings these kanji have; you'll get a feel for it from the kanji combinations you see them in ;) If you're interested in kanji, I suggest doing some reading about "radicals" which can help you guess the meaning of kanji you haven't seen before.
In Chinese 生 could mean person as well . So 学生 is actually a person who studies. And 先生 means Mr. or teacher（no matter male or female ). It's just a two-character-word . But maybe its oringin is what you explain. P.s. I only explained what these characters mean in Chinese, I'm not sure if they mean the same in Japanese.
I'm not sure what you mean by "giving two pronunciations", but I'm going to disagree with what the others have said so far. Stressed syllables are very subtle in Japanese, but they do exist. Perhaps the most well known example is the pronunciation of はし hashi, with HAshi (with a downward inflection) meaning "chopsticks" and haSHI (with an upward inflection) meaning "bridge".
In this exercise, I actually think the recording is incorrect. As far as I can tell, the correct pronunciation should be namaE with a slight upward inflection, whereas the recording sounds like naMAe causing the downward inflection. All in all, it doesn't affect the meaning but it doesn't sound natural. My guess is the audio generation software recognizes 名 and 前 separately and pronounces them accordingly (前 on its own is pronounced MAe with a downward inflection).
Also, the pronunciation of certain syllables does depend on the syllables around it. The most common one people have noticed on Duo so far is the kind of "crushed" su at the end of です and ます, causing them to sound like des and mas. Other common "crushed" syllables include し in して (sounds like shte) or つかう (pronounced tskau). I think there must be some kind of rule that describes how and when this happens, but I don't know it.
On stress: English, and most European languages, has a stress accent, which means that the accented (stressed) syllable is marked in three ways -- it is pronounced with a changed pitch, with a longer duration (speaking time) and often with a slightly higher volume. Japanese, on the other hand, has a pitch accent, which means that the accented syllable is marked only by the change in pitch. For a person used to stress accents, this can be hard to pick up, and is different enough that it is technicaly correct to say that Japanese does not have stress. It does, however, as you say, have (pitch) accent.
On "crushed" syllables: In most (but not all) dialects of Japanese, the vowels u and i are devoiced (or "whispered") when they are between voiceless consonants, as well as at the end of a word after a voiceless consonant (the voiceless consonants in Japanese are p, t, k, h, f, s, sh, ch, ts). However, if you have two such vowels in a row, the second is normal.
As the others say, although many (most) European tongues will want to put a stress on a particular syllable (like the second to last on in Spanish words and how to tell different words in English (see project)), Japanese has no such stress, and every character (mora) should be pronounced about the same length (mostly).
名前【なまえ】is a noun which means "name". When you introduce yourself using 名前, you usually say 私の名前は「name」です which means "my name is 「name」".
On the other hand, といいます is actually a sentence fragment made up of と, which is the quotative particle here, and the verb いいます (same as 言います) which means "to say" or "to be called". When you introduce yourself using といいます, you usually say 「name」といいます, which translates to "I am called 「name」" or "「name」is said (when talking about me)." (Except the verb here isn't actually in passive voice; it's the only natural-ish way to translate without resorting to imperative voice.)
Yes, family name or surname is 名字 (みょうじ) in Japanese. 名前 typically refers to either full name or only the first/given name. First/given name is also called 下の名前 (したのなまえ) for clarity.
Confusingly, on forms and other paperwork, surname is usually abbreviated to 姓 (せい), while first/given name is abbreviated to 名 (な).
That's what people in the evolutionary linguistics community would call an etymological impossibility, haha