[No 2. are both bad grammar. Do not use either of these.]
Persuasion can only be done before someone has decided.
"Why don't you take that class?"
This is often used to encourage someone to do something. It is more polite than telling them to do something.
They will respond as if it was a suggestion: "That's a good idea".
Questioning can only be done after someone has decided.
So they can respond as if it was a question.
To make it clearer that it is a question you can ask
"Why don't you want to take that class?" or "Why can't you take that class?"
If it is possible to change their minds then it is both a question and persuasion. If they don't like their decisions being questioned they will respond "Because I don't want to".
If it is too late then it can not possibly be a persuasion. It has to be a question by elimination.
When a question is equivalent to an imperative it is just as rude. Do not ask "Why don't you go to hell?" https://www.youtube.com/watch?v=jOflpNvy_30&t=0m23s
Aside from common names that you mention, native English speakers can be equally unfamiliar with names of people and places. If I hear a name I have never heard before I may double-check the pronunciation, or ask how it is spelled to help me visualise and remember that name.
There really are just so many proper nouns out there that nobody could be familiar with every one, and new names are created all the time. Even familiar names of people can often have multiple variations in spelling or pronunciation. The name Karen for example is fairly common in the UK and can also be spelled Caron, maybe other ways too. I have known people pronounce either of these as 'Kar·in' or 'Kair·on'. Some names used in English are borrowed from other languages and can sometimes be spelled in an Anglicised way but retain their original pronunciation. Names are meant to be individual, and people go out of their way to make even common ones sound a bit different.
Place names can be even more confusing, especially places within the UK and the US which have roots in either old English or native American respectively. Over time many places have changed spelling but retained something of the original pronunciation. Sometimes you really have to live in or near a place to know how it is pronounced! For example, I live near a small town called Wesham which is pronounced "Wes-ham" by people that live there, but "Wesh-um" by practically everybody else. Also, many British English speakers are baffled as to how US English speakers pronounce places such as "Maryland" (apparently Americans say 'Mari-lund', not 'Mary Land' as spelling and entymology would suggest) and so even if we have read a particular name we are not always fully prepared to recognise it when we hear it spoken for the first time.
So my answer to your first question of "do native speakers of English ALWAYS understand names they've heard for the first time in their lives on the first try" has to be no we don't! However, it is probably easier for us to know when we are hearing an unfamiliar name as opposed to an unfamiliar word because the likelihood of us not knowing a word is far less likely than an English learner. We probably also grasp context a little easier and are processing our understanding in real time as opposed to constantly translating in our heads.
As to your second question of "is something I can do to bypass this problem?" - all I can suggest is that this will hopefully become easier for you with time and practice. The more familiar you are with spoken English and the different ways native speakers can pronounce common words, the more likely you are to recognise that a proper noun is being introduced. Friends of mine whose first language is not English have commented to the effect that their learning English at school did not fully prepare them for speaking it with native speakers; however after time here they have learned to understand idioms, colloquialisms, accents etc.
One interesting thing I read about the way children pick up their native language is that they learn a great deal by what some experts call "chunking" - that is they learn "chunks" of speech first, then what the individual words mean later. For example, very young children may learn to say "good morning" as a phrase, not yet appreciating that it is two words "good" and "morning". It makes sense that this is the brain's natural way of learning language. But learning a second language is quite the opposite technique. You learn the words, the rules, and it is only when you start speaking it with natives that you learn how these are commonly strung together. I imagine that as you become more familiar with "chunks" of speech you will also more easily recognise the cue for a name to be inserted into the sentence.
Best Answer
If you say, "I come from India," it means that India is your homeland - probably where you were born and grew up. You use the present tense to denote that you are currently, and always have been, from India. The emphasis is on the continued state of being from India.
If you say, "I came from India," it means you traveled from India. You use the past tense to denote that the traveling is complete. The emphasis is on the past action of traveling from India.
But, as @SoronelHaetir said, if someone specifically asks, "Where are you from?" you can simply say "India" and the rest of the meaning is implied.