Yes, roughly so.
1) The major part of C to avoid is raw pointers and malloc, and generally anything where C++ has a more modern alternative. Don't use C datastructures (pointers, strings, arrays) where C++ replacements exist (smart pointers, std:string, std::array and std::vector, etc), or use C libraries where C++ replacements exist (e.g. C++ std::string vs C's string.h).
Of course you can't avoid all of C, since the syntax of C++ is an extension of C.
2) I wouldn't characterize what C++ adds to C as OOP. OOP is a design methodology, whereas the main addition C++ gives you is classes which are better thought of just as powerful and convenient type of data structure.
The core feature of classes you always want to use are constructors and destructors. Any initialization of the class's data members should be done in the constructor (don't leave things uninitialized), and any final cleanup (releasing memory or locks, closing files, etc) should be put in the destructor so that it always happens (even if your program throws exceptions).
Don't feel that just because classes support subclassing, polymorphism (virtual methods), etc, that you should be using them. They are there in case you need them, but if in doubt don't.
3a) The STL is just the standard library for C++. You should always be using STL data structures (std::string, std::list, std::vector, std::map, etc) when applicable - not just "for productivity".
3b) Templates (template classes, constructors/methods, functions) are not needed for most everyday use of C++. They are there for library writers, and on occasion for writing your own class and libraries. Think of them a bit like class inheritence - they are there in case you need them, but not something you should be reaching for unless there is no better way.
4) C++ has a LOT of stuff (esp. libraries) that might be considered as "for special case use only". A rookie mistake might be to think that C++ has all this stuff - I should be using it! In general you should ignore the obscure stuff, and only use it when some special circumstance requires it.