Learning C++ you will not know C. At least, not in the time it could have taken you to learn C by itself. There are a few core differences, and with C++ you'll spend a lot of time learning things that will distract you from the core language. If you want to learn C and understand it's advantages don't assume you'll pick up what you need by learning C++.
Yes you will. It's better to learn C++ straight up, because the things you learn there will help you become a better modern developer compared to just C. The core differences between C and C++ (apart from the entire object oriented thing) are easy for any competent developer to pick up. Were they difficult for you to understand?
The stylistic differences are wide. Learning to for instance do C style APIs using opaque types, extended macro-fu, broader use of function pointers, etc. are the sorts of non-obvious things that good C developers do that you'll mostly miss working just on C++ code-bases.
I don't think that non-obvious and "good developers" should be joined together in one sentence. It's possible to do a lot of macro magic, it's possible to do a lot of function manipulation, but this is not good development. This is efficient development when you are working with projects or systems that require that sort of thing (maybe large dataset manipulation or embedded systems), but in my opinion, doing those things is bad programming.
The reason is simple: They are difficult to understand. Code is communication, and there are two listeners - and when you are pushing function pointers around using macros, the human is going to have huge problems understanding what you are trying to achieve.
So I don't think exploiting particular features of a language to extreme levels when it is unneccesary makes you a good programmer in that language - I actually think it makes you worse.
It's like writing a book - you have a certain vocabulary at your disposal, and you can say almost anything you want with that vocabulary. There are different levels of obscurity in the words you use you can reach - you gain in expressiveness, but you narrow your audience. When you write a book for molecular biologists, then the particular terms you use should not be the same as when you write a biology textbook.
That's why I think that one can be a good C programmer without going much deeper than the C subset commonly used in C++.
This is part of my general philosophy in programming - clarity, simplicity and structure first, efficiency second. The machines will catch sooner or later, anyways.
It's not that those features aren't exposed in C++, it's that C++ has different (more modern) mechanisms for doing the same thing.
C++ has templates, so you don't need to do as much with macros. C++ has virtual functions and overloaded operators (allowing for functors), so you don't need to do as much with function pointers. Structures can have methods in C++, so the way that you'd set up an opaque type is quite different in C++.
C++ also has about 10000 other features as well though, which is what makes it unwieldy to learn. The patterns mentioned above aren't about efficiency; they're about encapsulation.
Just to give an example -- in C++ if you want to set up a callback, for example with an observer pattern, you'd use a pure virtual function (and then pass in a concrete instance of that class) or a functor. You can't do that in C, so you use a function pointer. You learn to work with function pointers more because they're a necessary part of designing C APIs, whereas they're not in C++.
(Note that it's the language I'm best at, followed by C, so this isn't random C++ hating.)
It's not about knowing the core differences, it's that the actual practice of writing a lot of C makes one familiar with its advantages. It's difficult to appreciate the value of C's small size if you are only coding in C++. Without doing a lot of coding in straight C, it'll be a lot harder to understand and recognize the various tricks and techniques other C programmers use to make the most of the small language.
I'm not saying there are no good reasons to learn C++. Just don't think that you'll really appreciate C until you've actually used it all by itself (with maybe a few C++isms here and there like //comments).