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
This would be a perfectly ordinary thing for a native English speaker to say. If you asked them to write down what they had just said, they would almost certainly write, "What are you thinking about?" even if they had not clearly articulated the words "what are" in their spoken conversation; they would simply think of this as the normal way to pronounce the words "what are" at normal, informal, conversational speeds. However, an author (for example of a comic) might want to evoke their character's conversational style, or even dialect, with spellings or contractions that reflect how the character is actually speaking, so that the reader will be able to "hear" the conversation more like the author intends. So, writing "What're you doin'?" gives you, the reader, a feeling for how that character's speech would actually sound.
For years (decades, centuries?) we have been taught in school that it is grammatically incorrect to use a "dangling preposition", and that "correct" sentences should always keep prepositional phrases together, such as, "About what are you thinking?" This is certainly not the rule in ordinary conversation, and that idea is also in decline even for formal writing, but it still holds some influence in formal English, and in some contexts it's still a good idea to try to keep the preposition and its object together if the result isn't too awkward. In ordinary conversation, though, that's definitely not a rule about which we usually think... er, I mean, that's not something we usually think about.